Vlad is really weak in the early levels because his Q doesn't deal much damage, has low healing, and is on a long cooldown. Combined with the fact that his W and E cost health, he can have trouble fighting in the early game. Many assassins or bruisers with a method of gap-closing onto him can typically beat him in a straight up fight around level 2 or 3, provided they haven't been poked too low first.
Vlad can't really stop a champion from pushing him under tower, which allows you to harass him while he's using his abilities to kill minions. If you keep him low, you can zone him from the minion wave and prevent him from healing up.
After Vlad gets some spellvamp and levels, he will probably attempt to push you under tower using his E and harass you whenever you try to CS. The tower isn't a big obstacle to him since he can heal up a tower shot's damage while using his spells on you. If he's pushing like this, call your jungler since Vlad doesn't really have a great escape (pool is nice for dodging abilities, but the MS won't really let you get back to tower). Try to avoid harass while CSing and keep your health up, since Vlad can easily burst a champion who's low with his ultimate and ignite.
Vlad is primarily sustained damage and doesn't have great burst damage unless he's ahead. If you can all-in him before you get poked low, many champions have an advantage against him. Though, be careful about using important spells until he has used his pool so you don't waste them.
Ahri and Orianna are pretty good champions to use against him since they have low cooldowns and good waveclear. You have to watch your mana early game, but you should be able to outtrade him using Orianna's passive or Ahri's Q harass and zone him from minions if you play aggressively.