I am going to divide the path into larger categories with explanations and step by step instructions. I will summarize the path initially without explanation in order to give the reader a quick reference. Execute the path in the order that it appears in the summary(top: first; bottom: last).
(1) Blue Buff
(2) Red Buff
(1) Blue Buff
A) Wolves: Your team needs to auto attack these so that you can clear them in time to start blue. It is ideal for you to draw the agro yourself, since wolves run quite fast, meaning that you can waste time trying to chase them down. If your team does not give you any help, simply clear the large wolf and come back to this camp after blue.
B) Blue: Allow your team to leash this for you. If your team does not attack the buff several times, it could require a second drain. Use smite to last hit the blue. This prevents blue from being stolen by an enemy smite. Skill dark wind (E) as soon as the blue dies, and use it on the small creeps to clear them more quickly. You may want to activate your potion during this fight if your team did not give you a very good leash(or no leash at all).
Starting at blue is necessary as fiddle, since buffs are much more valuable than basic camps in terms of exp and fiddle is extremely mana dependent. In order to optimize efficiency, you have to clear the wolves before you start blue. This will generally require you to communicate with your team and instruct them to help you with wolves and then leash blue(At higher elo, this can be done with a simple ping. At lower elo, you may have to type in chat.). You will get counterjungled at blue from time to time. I will summarize how to deal with that later in this section.
(2) Red Buff
A) Red: Lead with E and stand in the bush near red while you drain. I like to pull the red entirely into the bush, so that my opposing jungler will have a harder time spotting red if he/she comes through while I am taking it. If you do red immediately after blue, you will not have smite. But smite is unnecessary if you started with an hp potion and have correct runes. You may decide to skill Q and gank at this point.
B) Golems: You will hit level 3 here.
C) Red: In this case, smite will come off cooldown in time for red and you will be level 3, so your red is a bit safer in those regards.
I suggest taking red first because, even though you will not have smite, the timing at which you would take red is really unconventional, and is thus protected from counter jungling(ie. no conventional counter jungling path would put your opponent at your red at that time). Furthermore, red will put you to level 3 and allow you to skill fear (Q) and gank immediately afterwards. This allows for a very fast level 3 gank with red and blue while still being the fastest way to farm exp. Option 2 allows you to take red more safely, with 2 points in drain and smite off cooldown. If you don't have the right runes, Option 2 might be preferable.
You will have to option to gank after each camp once you have put one skill point into fear(Q). You have the option to skill fear at level 3, and you need to decide whether or not do so while you are taking the camp that will get you level 3. To do this, you need to be aware of the lanes. Watch the laning that is taking place in nearby lanes, and decide whether or not a ganking is possible.
I will state the obvious here: you need to strike a balance between jungling and ganking. Generally, you should not gank while you have a buff up, since buffs are extremely valuable, although this is not always true. Because basic camps become more valuable over time, it is usually okay to let the 3 basic camps sit there while you gank knowing that you can clear them afterwards.
There is a lot of variation that goes into a solid jungle fiddle style. As such, in order to jungle fiddle properly, you need to be acutely aware of the status of all of your lanes. If you don't trust your map awareness or your ability to judge whether or not a gank will be successful, I encourage you to simply rely on the path that I have prescribed. But keep in mind that in order to jungle fiddle at an extremely high level, you will have to develop a good sense of when you should gank.
Tips on pre-6 ganking execution(copied from comments): WALK TOWARDS YOUR OPPONENT'S RETREAT PATH, not towards your opponent. (Walking towards them would be like sending a pass to a retriever's current location, rather than towards where they are going). Lead with fear, use auto attacks(assuming you have red buff) and dark wind while you position yourself DIRECTLY IN YOUR OPPONENT'S RETREAT PATH (ie. move, attack, move, attack, move; this takes a little bit of mechanical precision). Then use drain when you are DIRECTLY IN YOUR OPPONENT'S RETREAT PATH. If you couldn't get great positioning, you can drain them as they are getting away just for the hell of it. Also, PING PING PING PING PING PING PING PING PING PING PING PING PING to alert your allies. Make sure your allies are capable of going aggressive before you move(ie. there is not a big snowball of creeps for your opponents).
Dealing with the Counter Jungle
Fiddle is a blue dependent jungler. Thus counter jungling can really shut him down. I will give a few suggestions about how to deal with counter jungling:
1) Ask your teammates to scout, so you know if your opponents are coming in.
2) If you feel like the opposing team has a strong level 1, you can use your ward on the small brush in river near your blue. This will let you know if your opponents are coming without putting your teammates in danger.
3) If your opponents come in and you feel like your team is capable of fighting them off, you can use your ward on your own blue brush. Vision of that brush will make it very easy to force your opponents to retreat.
4) If you feel like you can defend your blue, but are worried that your opponents could steal your red, you can ward your red. Alternatively, you can ward an entrance to your jungle that an opponent might take on the way to your red. I have afforded my team first blood this way in high elo.
5) If your opponents invade your blue and there is absolutely no way to scare them off, you have three new starting options: (A)their blue, (B)your red, and (C)their red. Decide quickly which one you will do:
5A) Usually they will expect you to try their blue, but often times they cannot stop you if they are stacked at your blue. Ask your team for help if you go for their blue, but remember that it is dangerous and, so be ready to flash away if your opponents come.
5B) Its is very safe to start at your own red in this case, but of course, this is the least rewarding action, since your opponents would probably not have tried your red anyway, so you aren't taking anything away from them.
5C) Your opponents will not expect this at all, but you have to be sneaky, and you should not do this unless you are sure that your opponents are staying for your blue. If you do this, base immediately and sprint towards your own red so that it does not get stolen. Remember that your team can give away your position on this one if they are careless, so encourage them to be careful.
If you went with B or C, you need to get 2-3 mana potions when you go back to spawn to sustain your jungling. Extra mana is costly, but you will need the sustain in order to jungle effectively.
6) Do not panic. This is paramount. Remember that when your opponents do steal your blue together, they are sacrificing lane exp in order to do so. Don't think that you are losing horribly if you lose blue. Just relax and think clearly about how you can farm exp as efficiently as possible.
~The ability to play through a disadvantage is essential for playing fiddlesticks well.~
GET A PINK WARD AND PLACE IT DIRECTLY AT THE ENTRANCE TO DRAGON BEFORE ATTEMPTING DRAGON
Dragon and nash are a huge part of fiddle's potential. He can solo both of them extremely effectively in comparison to most other champions. So lets start with the dragon:
Fiddle can solo dragon starting at level 5 with rank 3 drain and a blue buff. Whenever you take dragon, you want to make sure that you take it from an optimal position. That usually means that you need to leash it out of its cove before you start spamming drain. Having good positioning while you are taking dragon will allow you to retreat if enemies come to terror you.
So there are 4 positions from which you can take dragon:
Position 1: Only use this position when you are on the top right(aka purple) team. If enemies come from either mid or bot, you can simply run back up the ramp towards your blue camp. This is generally a really safe position, and is the reason why purple side has an advantage on dragon.
Position 2: Only use this position when you are on the bottom left(aka blue) team AND HAVE FLASH UP. You would use this position if both mid and bot are pushed for the enemies, but you do not think that your enemies have not warded the dragon yet (maybe because they haven't based or because it doesn't seem like they bought a ward). If enemies come, you can simply flash over the wall towards your red jungle. Be careful if you see both lanes leave immediately though, as your opponent from mid might know what you are doing and simply go behind the dragon cove to terror you after you flash over. Sometimes it is worth it to just flash right away if both mid and bot enemies react immediately. If you take dragon from this position, you are basically betting your flash that your opponents did not ward dragon yet. A lot of times, that bet will pay off.
Position 3: You can use this position when you are on either team, although if you are on the top right(aka purple) team, you can position yourself farther from the wall to put yourself closer to your retreat path. You can use this position if your ally is pushed up in mid(thus preventing mid lane enemies from coming to terror you). If enemies come, Use the retreat path towards wraiths if you are on the blue team, otherwise use the northward retreat path.
Position 4: Only use this position when you are on the blue team. You can use this position if your allies are pushed up in bot lane(thus preventing bot lane enemies from coming to terror you). If your mid lane enemy comes, you can simply take the retreat path towards down your triangle bush.
If you have allies helping you with dragon, stand directly next to it so that you are tanking it.
Fiddle is really good at taking nash. If you have a deathcap and a blue buff, then your team is capable of taking nash as 5. If you are level 12 with a deathcap and blue buff, then you can 2 man nash with any sustainable or tanky champion on your team(to list a few: nidalee, soraka, vladimir, any tank in the entire game). If you are going to 2 man nash, try to bring a damage dealing sustainable champion, as that will allow you to take nash more quickly. Also make sure to tank nash for your ally most of the time, only letting your ally take damage when you go below half health. Nash selects his target based on who is closest to him, so you need to make sure to drain from "inside" of him. You can do this by clicking to move on the other side of him, then starting the drain when your are right on top of him. I would definitely encourage you to try soloing/2-manning nash as much as you can, so that you can learn to judge when it is a possibility. You will almost always need to have a blue buff to do this. Generally though, you should try to kill some opponents first and then take nash with your team.