Yi is super squishy usually so when playing with my low level friends I was playing around with Yi and found that Tanky DPS Yi can carry pretty hardcore (assuming they don't have too much CC).
I'm making this specifically for one of my friends so I don't really care how this is rated and I don't want to put too much time into it. I also haven't tried this at 30 with my current elo so I don't know how well it will do there, but I never play Yi in the 30s so whatever.

Approximate end game stats (this was done without runes and includes damage from Atmas):
<3000 health
~180 armor
~120 magic resist
~280 damage per hit
~1.6 attack speed
~450 move speed

Changelog:
- fixed some coloring issues
- fixed some paragraphing issues in Summoner Spells + Masteries + Runes
- Changed the summoner spells to include flash instead of ignite for the "proper" build
- Added a brief Team Fights section
- Added in approximate stats
- Added more detailed item section

For non-30 players:
Focus for masteries should be:
Deadliness: 3
Cripple: 1
Alacrity: 4
Sunder: 3
Sorcery: 1
Brute Force: 3
Sorcery: 4
Archmage: 1
Havoc: 1
--
Perserverence: 3
Good Hands: 1
Awareness: 4
Utility: 1
==================

Summoner Spells:
Exhaust: Important to prevent escapes and add extra dps with the Exhaust mastery
Teleport: Good for backdooring late game and getting ganks by teleporting to wards.


Ignite: Decent to finish people at towers early or cancel high hp regen people late game
Ghost: Viable but not really needed because of Yi's Ult
Cleanse: Not a bad choice if they have lots of CC but almost impossible to know how much they will have. Can save your life and help your dps output if they do try to CC you
Flash: Helpful in some situations, I feel that other summoners are better suited for yi
Clairvoyance: There are better skills for Yi


Fortify: I never use it, don't see a use especially for Yi
Heal: No. Never.
Rally: Not really a good spell
Revive: No. Never.
Smite: Not jungling.

======================
Runes:
HP Quints
Armor pen reds
Armor/dodge yellows
mana regen/magic resist blues

Item build order and reasoning(including some variations)
On game start:
1. Boots of Speed- dodge projectiles and move fast so you're scary
2. 3x Health Potion - so you don't die when you're being scary
3. Vampiric Scepter- life back per hit, necessity for Yi
3a. Berserker's Greaves- attack speed, faster movement. Everything Yi wants
4. Phage - slow them so they can't get away, more survivability, some damage
5. Zeal - better chance to slow on phage ( more attacks per second ) some crit and move speed, what's not to like

Now is where it may vary.
Pick one than the other:

6a. Frozen Mallet- Lots more life, very little extra damage, Always slow on hit 40%, Very nice for ganks and to prevent kiting.
6b. Phantom Dancer - Lots of extra attack speed, lots of extra crit, doubles zeal's movespeed boost. Good for more dps than Frozen Mallet will give.

Now, if you need armor and more damage per hit:

7a. Atma's Impaler - By now you should have around 2200-2500 health, so if they have lots of physical damage that's killing you this will give you +45 armor, 18% crit and an extra 44-50 damage (increasing as your health goes up)
7b. Banshee's Veil- If they have lots of magic damage or stuns that are killing you, this should be next since it will give you more life (to stack with Atma's in a bit) magic resist, and a shield to block a stun (first spell on you hopefully)

Final Item Options:

8a. Bloodthirster - Lots of extra damage (caps at +100 total), up to 25% life steal. Need I say more? If you feel like each hit isn't doing enough, this is the best option
8b. Stark's Fervor - Lots of Attack speed, health regen, more life steal, and debuff on enemies for -20 armor. All around really good for you and your team
8c. Wriggle's Lantern - Attack damage, Life Steal, more armor, and a reusable ward, great choice as well, though it should be gotten earlier in the game, probably prior to Atma's or Banshee's

Possible Item switches:

If they stack armor a lot it may be necessary for you to not get the Frozen Mallet/Atma'scombo and instead get a Last Whisperfor the 40% armor pen and 40 attack damage, as well as a Guardian Angelfor the armor, magic resist, and resurrect passive since you will be getting rid of 700 life or a Sunfire Capefor the armor and health and the passive damage around yourself.

If they stack a lot of life, getting a Madred's Bloodrazorand get one of the previously mentioned items as well, still getting rid of the Frozen Malletand Atma's

Start with the boots of speed and 3 hp pots.
Make sure you get some good farm so when you b you can get boots and vamp scept. (Maybe part of Phage such as the hp crystal since Yi is super squishy early game)
If possible lane with someone who can help feed you kills so you can get items faster.
Since the build does require a lot of money to get completed kills and farming are necessary. However don't be overly aggressive trying to get last hits since you are squishy it will be easy for them to gank you. Use Alpha Strike to your advantage and remember to watch out for ganks.

Yi is pretty weak early game so don't be discouraged if you don't have a lot of kills and maybe even a few deaths. That doesn't mean it's okay to feed, it just means you're not screwed if you don't have a baller early game.

After you're around 9 or 10, it's about the time when Yi starts to pick up. You should have at least the Phage built if not Zeal already. Now you should be able to gank and break even on kdr if you have been killed a couple times (and hopefully have assists at least equal to those deaths). You're still not a big fat baller so don't run in 3v1 or anything but you can dish out some good dps and chase people down with the slow on Phage and the attack speed boost from the ult and Zeal.

Thought Yi isn't really a caster, if nobody on your team NEEDS blue, you should take it since Yi does like to eat all of his mana pretty fast. Getting double buffed (red and blue) makes Yi a very scary person especially when he starts ramping up.

Once your ult is level 3 and you have Atmas, (You may switch Banshee's Veil and Atmas if you need more damage faster and aren't in dire need of the magic resist and 375 hp) you should be super tanky dps man and be able to dominate in team fights or 1v1 encouters (assuming they don't have tons of CC and/or aren't super fed). This will allow you to finish any other items you need and pot up and win (if you haven't already).

Since this is a tanky dps yi build, you don't have to wait til the fight already started for you to go in. That doesn't mean run in before the tank or with the tank, but with the rest of your team, maybe a little bit before the non-tanks. Just try to avoid stuns because you still rely pretty heavily on lifesteal to stay alive.
Make sure you focus their squishy carries (usually the ranged carries first but make sure to target the scariest person first), you'll rip that character to shreds then move on to the next scariest person.

The pros for this build:
- More survivability than most Yi Builds (Especially while meditating)
- With the more survivability it allows Yi do dish out his amazingly high damage longer, especially if he can use his ult several times in a team fight.
However, this build does have one major con
- Lower damage than many Yi builds

This is just another look on how to play a cheap champion and do well, especially while leveling up.