I main as an AP player in around 2k ELO, so I've had my share of Vladimirs past 2 years!
Laning vs. him:
Vladimir is actually rather weak early game due to the fact that his Q (Transfusion) is on a high cooldown and doesn't heal for much. If you're playing offensive AP champs who are strong early (Cassiopeia, Ryze, Ziggs, Kennen) you can punish him by going aggressive. Also make sure you exploit his short range on his Q.
Playing vs. him during teamfights/in general:
Since he is a short-ranged caster, you can avoid most of his damage with good positioning; make sure you stay behind your tanks.
Also, if he's really fed you can go for Quicksilver Sash to cleanse his ultimate off, which shuts down a lot of his damage.
Heal's debuff helps a lot too, since he has quite a lot of sustain, so make sure to grab that, or at least try to make sure the AP on your team has it.
Hopefully that helped, good luck!