Ok, so I looked up your match history, and your runes and masteries check out with whom you're matched-up against. But... I feel there are a few things that went wrong.
1: You didn't get Q at lvl 2. You should use your q-e combo at lvl 2, because with flask, you can regenerate mana fairly easy at lower levels, and at lvl 3, your power-spike is stronger than vlads (i believe.... may need to test)
2: You maxed E, I normally either max w or q depending on match-up.
3: After you died, you kept dying. Something to keep in mind playing as Jax, you need farm really bad. If you feed, your Jungler is NOT going to gank 4 u. so farming some jungle camps or roam is better than trying to farm and failing and dying. Don't go back to lane to die.
4: You shouldn't die under tower. Let me explain. If you're just trying to farm under turret, try and bring minions further back to be able to farm safer. If vlad dives you, then he'll trigger the turret. In a case in which you're both 100% health, Q-W to him, and if he uses his W, use your E to stun after he gets outs of troll pool. Then walk back to farming unless you have the upper edge and he's not running away. If you're too low, recall after you clear waves, and he has a 26 cd on w, while you have a 16 CD on E. He'll die if he dives you again within that threshold of time.
5: Speaking of Cooldowns, he becomes untargetable for two seconds, and your E dodge lasts TWO seconds. Basically, use your E very shortly after he W's, and you'll have a near definite stun.
Let me know if you disagree on something, I'd love to chat some more. Or let's play a game together sometime.
EDIT: I'm Talking about the match-up between Jax and Vladimir if anyone happens to get lost.