Although YI's abilities aren't too complicated, there are many ways to make use of his damage.
Basically what you want to do is getting your Lich Bane as soon as possible and then stacking AP, you can use any spell to deal damage with Lich Bane to finish enemies for resetting your ult.
Why would you pick Yi and not any stronger laner with more damage abilities?
You would pick Yi vs. squishy teams that lack tanks, for example: a Fiddlesticks jungle, Lulu support and a Katarina top, carries aren't able to stack big amounts of mres, which is the only way to make Yi useless since he can't 1-hit players then. If you pick Yi vs. tanky teams, you will have to wait for your team to get them low, then just clean the floor with your ult resets.
In teamfights: don't use your heal to tank anything, use the heal and your ult to save yourself, don't go too deep. Try to damage every member of the enemy team, kill the weakest targets first while you get the stronger champions lower and finally get them all.

Pros: destroys carries, strong laning phase vs. champions that can't deal with his meditate, good poke in teamfights when used right
Cons: needs a big amount of manaregen (laning) as well as flat mana (teamfights), is screwed against magic resist stackers, such as galio or Ryze, don't let them counter you

Dorans Ring: really good item, you can heal yourself due to increased mana reg, Q scales with 100% AP, although i feel like legendary items are more important
Zhonyas: better choice that GA if you need the damage
Void Staff: good damage boost and generally good item
Abyssal Scepter: awesome item to destroy carries with only 1 Q + hit
Seraphs Embrace: you don't need that much mana, but: the shield is amazing and the AP boost is huge
Rod of Ages: Yi doesn't need health at all, he's an assassin
Liandrys: excellent item imo, early mpen works very well since his Q already has a big base damage
Mejai's: he doesn't have the range that Orianna or Lux have in order to stack AP without dying, get it if you're really far ahead maybe
Iceborn Gauntlet: he doesn't have AD at all except for the small AD buff from his E, everything else is fine with this item
Morellonomicon: should work, gives huge CDR, but Ignite will probably do enough if you have problems with a certain target

Flash: you need this to follow champions for ult resets
Ignite: you don't want players to get away from you with 100 health when you could have had a Penta after the ult reset
Teleport: his ganks are decent, very awesome spell
Exhaust: use it with the mastery to deal more damage
Ghost: the only situation where you could need this is in teamfights, but people will throw all their CC on you as soon as they can click at you
Barrier: useful, but not as epic
Heal: useless
Revive: Revive Teleport should actually work very well, try it out
Clarity: Yi is one of the few champions that can make use of this spell in teamfights
Cleanse: vs. heavy CC teams, to continue Pentakilling