Trinity Force - Trinity Force is good on Nocturne because pretty much every single stat is used and scaled well.
Attack Damage - Helps deal more damage to secure kills. Nocturne's AD ratios aren't the best, but they aren't the worst either so this helps increase the damage of Duskbringer and Paranoia quite a bit.
Attack Speed - Nocturne has a natural attack speed steroid from Shroud of Darkness ( W ) which is doubled when a spell is shielded. Attack speed scales with itself, meaning the more attack speed you have, the more you receive from bonuses. Since Nocturne already has an attack speed steroid from his W, the attack speed from Trinity Force scales really well.
Critical Chance Strike - This is just a luck stat. 10% chance to do 200% damage from auto attacks. Since Nocturne is an auto attack based champion, this does really well with him for surprise damage whenever you get a lucky critical strike.
Movement Speed - The movement speed bonus from Triforce works so well with Nocturne if you add it along with Duskbringer, runes, and masteries. Having movement speed bonuses from Triforce, runes, masteries, and Duskbringer gives amazing chasing potential which helps secure kills. Nocturne already has strong base damage from the bonus AD from Duskbringer and the attack speed from Shroud of Darkness, so being able to stick onto your enemies to constantly apply your damage is important. If you tag an enemy with Duskbringer, you can just follow the trail, and in no-time you will be able to catch up to enemies. Having redbuff to slow your enemies makes it almost impossible for them to run away which can either result in a kill or a flash burned. I've found multiple times that while split-pushing, I was able to run away because of how fast I was whenever I was being ganked. Having multiple enemies trying to chase you down can give your teammates the chance to take objectives such as towers or dragon.
Ability Power - This is probably one of the most useless stats on Triforce for Nocturne, but it still isn't THAT bad. Nocturne's Unspeakable Horror ( E ) has a 1.0 ratio which is very good.
Health - Health is always nice. Helps you tank more damage to prevent yourself from dying. Simple.
Mana - Nocturne isn't too mana thirsty, but I've found times that I've been almost out of mana after teamfights which the extra mana prevented me from going completely out of mana.
Unique Passive - Rage - This passive gives EXTREME amounts of chase potential. Since Nocturne is an auto-attack based champion, this passive is almost constantly applying during fights. With this movement speed along with the movement speed from the base stats of Triforce and the bonuses from Duskbringer, enemies can almost never outrun you even with ghost. This makes applying Unspeakable Horror extremely easy. Nocturne has high consistent damage because of the bonuses from Duskbringer, so being able to stay close to your enemies to apply the damage gives great kill potential.
Unique Passive - Spellblade - Nocturne does not have many spammable spells to constantly apply this. However, this passive makes it really easy to kill squishies such as the enemy Marksmen or AP Carry because of the crazy initial burst. Duskbringer along with Umbra Blades and Spellblade creates a massive burst that people do not expect. Spellblade is not as good on Nocturne as let's say an Ezreal or Irelia, but it is still really good.
Other options for attack damage is Hydra, Blade of the Ruined King, or Bloodthirster.
Hydra - Hydra gives massive amounts of attack damage, but it does not give attack speed which I think Nocturne can benefit well from because as stated earlier, attack speed scales with itself.
Blade of the Ruined King - So this item gives about the same amount of attack damage and attack speed, but it doesn't give the burst from Trinity Force. Blade of the Ruined King has an active, but it can only be used once every 90 seconds. I also think the Movement Speed bonuses from Triforce is better than the Movement Speed bonus from the BotRK active.
Bloodthirster - I don't use Bloodthirster for the same reason as why I don't use Hydra. BT does not give attack speed, and also it needs to be stacked.
Life Steal is a good stat for long fights to sustain your health. You don't want long fights. You want to assassinate the enemy carries before their team can come and help them. If you're teamfighting, chances are you won't be able to constantly auto attack the enemy carries long enough to heal your health back up. If you're looking for more health, buying flat health is the way to go.