I will try to include most of the relevant match ups that you will meet and how to act/beat them, if you feel some are missing feel free to ask.
I will go through the champions alphabetically and add a difficulty score (1-10) where 10 is the hardest and 1 is the easiest.
You will find it nearly impossible to kill him. Once he gets Wardens mail you basically can not touch him unless you have a good advantage. However, you should be more or less free to farm, build some early mr/regen to mitigate his herass and you should be fine. If you are well ahead you can go in and try to combo him, remember to have ignite ready so he doesnt kill creeps with his AOE and heal up midfight.
A really, really hard matchup. It's even hard to farm in this matchup. Even if you are ahead of him item-wize you can still lose trades and get killed by him. So be very afraid when against Darius. Your best bet is to get your jungler to come help you out, Darius biggest weakness is that he is very, very, very easy to gank, since he will auto-push the lane when he goes in for herass with his Q. And since you're Irelia you can easily put him in a bad situation when you start off by stunning him with your E and letting your jungler get close.
It is also very important to get Exhaust in this matchup, you can actually beat him if you manage to get him to waste his Q, then stun him and proceed to fight. Near the end of the fight put exhaust on him to mitigate the last of his burst and hopefully kill him. Exhaust also makes it easier to gank him.
She is a good herasser in lane and can be difficult at times, but you should be able to win overall trades, take advantage of her lack of ultimate and go at her from level 6-ish. Bring as much sustain as you can to the lane.
Matchup is fairly easy, he does mostly physical damage and you should be able to out-damage him when he spins by simply atttacking him with your W. Not much to say.
A hard matchup, but it depends a lot on how good jax is. Basically the one who initiates a trade is the one who will lose the trade - He will proc Counter attack as soon as he hears your W activate and you will be unable to do most of your damage on him. However you can out sustain him in the long run with simple and quick herass - dont do long all-ins and farm as much as possible.
Good herass and can be hard to deal with. He is similar to Elise where he lacks an ultimate spell, you can take advantage of this if he tried to all-in you. You outscale him well, so just focus on farming and kill him with ganks - remember to dodge his Q's. If you want to kill him, you should gapclose to a creep, then stun him and hit - when he knocks you away you should gapclose to him with Q and proceed.
When you fight, always fight in range of creeps - dont let him lure you away from them and basically outsustain him. You win straight up fights with your W and you should be able to do quite well against him most of the game. Build some armour to mitigate all of his damage and you should do well.
One of the hardest matchups actually. If he maxes his E you will be unable to do ANY damage against him, because he will lower your attack speed insanely and you are very reliant on attack speed to get off W hits. Basically just farm the best you can, Lee Sin gets worse as the game progresses and you outscale him insanely much.
You should be able to outsustain all of his herass, when you trade make sure to use your W after his attack speed debuff wears off, try to keep his shield down by poking him and basically just beware of ganks when he reaches 6 since they can have a crazy amount of CC to keep you down.
She has a bit of poke and is hard to kill. You generally shouldnt be able to kill her, and might want to take teleport as summoner spell 2. This way you can outfarm her and follow her when she tried to split push, and generally just outscale her. She is very weak before level 6, so if you want to take her down that's when you have the chance.
Most people find this a very hard matcup, but I've found that you can trade very well with him generally through the entire laning phase. Make sure to always stun him and get some hits off before he can do anything (pre-6) and after that just make sure you have the creep advantage and dont let him herass you too much with his E - punish him if he tried to as it has a long CD at first. The difference here is that his true damage, does damage to himself - while your true damage heals you. Ignite is also very important in this matchup as he heals a lot when he gets lower in hp.
A hard matchup, he will forcefully herass you through the entire laning phase and be very difficult to take down once he hits level 6. What I usually do is that I go for the Runepage 3(AR/AR/AR/AR) which enables you to outsustain basically all his damage and just stand still and attack him if he tried anything funny. Basically the key points are: Build some armour, farm the lane and outscale him lategame. Renek falls off immensely at the later stages of the game.
You usually do very well againts Riven. When she goes in for combo just proc your W and hit her back, stun her when you are at lower HP and keep hitting her. You win trades and outsustain her easily - if you want even more of an advantage go for the Runepage 3 and watch her cry herself to sleep. But beware that if you engage her, she can disengage you easily and if she engages you while your W is down, you're gonna have a bad time.
Build some magic resist with runes/masteries and perhaps get an early +MRitem and you should be mostly fine. Be careful that he doesnt kite you around but other than that - you should be able to win trades with your sustain and true damage and outscale him later in the game.
Can be hard to deal with sometimes, you can easily be ganked when they have a shen(because of his taunt) and you cant really kill him if he doesnt want you to. Best bet is to play pretty much the same as against nidalee, pick up Teleport and travel around with him for ganks, else just farm up and own him lategame.
You shouldnt pick Irelia against singed. It's that simple really, if he is a good singed player he will just make the lane unbearable for you. You cant kill him and he will just stand behind your tower and take all the creeps while you try and lasthit under your own tower for as long as you can. You have almost no pushing power to push the creeps back. Your best bet is to get your jungler(and mid laner as well) to gank him, CC him down and ignite him so he doesnt regen - and Kill him before it's too late..
Remember: Dont chase singed.
There is almost nothing you can do in this matchup after he reaches level 6. If you try and go for him he will just stand in the middle of the lane and heal up. Basically just farm (while avoiding his bind) and if you want to kill him, make sure that he is not close to any of your creeps and that you have ignite on him AND that you dodge his bind. Other than that, farm and outscale.
Fairly straight forward matchup. Gapclose him, stun/slow and just hit untill he pools then run away. Rince and repeat till his sustain runs out, ignite him when you plan to kill and you should be able to get away with it. Make sure you dont let him freefarm since he becomes a beast lategame.
Farmlane. None of you can kill the other - but be aware of his jungler, when WW reaches level 6 he will most likely ult you when his jungler is there, and they will try to combo you down. Other than that, just farm and outscale him.
A lot of physical damage, get some armour and dont be fooled by his invis. He wants to do quick pokes on you and go all-in once his ulti is up, basically just punish him and heal back up once he pokes. He falls of later in the game and if you play it right, you can even kill him on lane.
Same as wukong. Once he jumps you stun him, then W and just hit him, or if you feel like you are taking too much damage run around untill his Q+W is over, then engage on him with your Q+W.
Can be hard to deal with sometimes, but basically just sustain sustain sustain. Might want to get the health reg quints for this one - just farm and sustain the lane, he will eventually run out of mana if he spams you, you can dodge the ghouls by going into grass (if he loses vision of you the ghouls go to creeps). You might be able to all-in him at level 6 with ignite, never without ignite though. He is also easily gankable.
Might want to bring Exhaust for this one. He is an assassin with Herass, sustain his herass and if he all-ins you at level 6 just exhaust him and kill him. Not much to say other than watch out for his escapes and dont get hit too much by his E.