Yorick is probably your best choice.
I would recommend:
Masteries - 21/9/0
Runes - AD Marks, Armor Quints, MR Glyphs, Scaling Health Seals
Item Build: Crystalline Flask + Potions, Tear of the Goddess, Ninja Tabi, Manamune, Iceborn Gauntlet, Spirit Visage, Randuin's Omen
Skill Order: EQEWER, R>E>Q>W
Summoner Spells: [Ignite or Teleport], [Flash or Ghost]
When he is CSing early game, autoattack him when possible and use E on him during trades to heal. At level 2, use AA-Q-AA to harass him and save E for when you need to heal. You do not want to use W that often early because it costs quite a bit of mana. You don't necessarily need an early kill and you shouldn't waste mana for it, but harass him low enough that he misses CS and has trouble approaching the wave.
Very important, you DO NOT want to push the wave early because that exposes you to ganks and you don't have the mana to sustain ranged harass if he's under tower. You want to freeze the wave near your side and bully him away from CS while you last hit. Don't be afraid to use your ghouls or yourself to stop the minion wave from reaching your tower (you can heal with E after).
By your first back, you should have enough for Tear of the Goddess. If you have extra money, I would recommend buying Doran's Shield to survive if you are having trouble, Doran's Blade you are even and want damage, or Boots of Speed if you are ahead significantly and want to kite and chase Jax.
Later in the game, when Jax has some levels and items, you don't want to fight him straight up. He will likely outdamage you in a straightforward fight. You want to take advantage of your slows (W and Iceborn), movement speed (Q), and sustain (E) to draw out fights. Instead of standing nearby autoattacking him, use your ghouls on him, hit him with Q and Sheen, and then back away. You can draw out an engagement this way, giving his passive stacks time to drop, allowing you to heal, and letting your Q and E come off cooldown. If he ever tries to all-in you with a slow or his ultimate, drop your ghouls and run if at all possible.
If you want to watch someone play Yorick, this Youtube uploader has some good videos: https://www.youtube.com/user/3Ghouls1Yorick
The uploader also has a guide here: http://www.lolking.net/guides/284611