Out of all the armies available I have played against the vamps the most and have had the worst luck against them.

So I will just give you a quick overview of what units have experienced some degree of success against them:
1. Witch Hunter - For his MR. I gave up on trying to kill the vamp lord with him, instead naming the BSB or a necromancer as his primary quarry. They die easier and are worth more than this over-fluffed under strength hero so he can at least make up for his point cost.
2. Peg Cap - Give him a berserker sword and a dawnstone along with all possible armour upgrades and he becomes the most successful unit. He can quickly remove any flankers or redirectors and does a pretty good job at bogging down bigger nastier units, like blackcoaches or varghulfs, for a turn or two. (He once killed a varghulf and bogged a blackcoach down until the end of turn 4)
3. Halberds - 40 or more Halberds for the win. Against low BS vamp chaff, compared to elves and others, swordsmen are a waste unless you really want a parry save. Buff them with a BSB and an AL or priest and you have a unit that will practically never retreat and will be able to dish out some damage against the stronger units like graveguards. Add 2 detachments of halberds that benefit from their steadfast and other rules to bog down any flankers or be flankers themselves. If you have a very mobile army that needs a good anvil this is it. Swordsmen could possibly also do but they cost more and their only task is not dying to quickly therefore I go for the cheaper +1 Strength Halberds.
4. Helblasters - With an engineer you are bound to take something big down before a serious misfire or the battle becomes melee only.
5. Canon - do not underestimate the power of strength 10! Blackcoaches and vamp lords alike are all chaff before this monster. Out of all the games that I managed to kill the vamp lord it has always been this baby that did the trick.
6. Lore of Heavens - Though there may be other lores with better buff spells or better damage spells the Wind Blast of this lore is a good companion to any canon trying to snipe the vamp lord. One of the games I managed to win I kept on pushing the vamp lord and his unit out of charge range until round 5 when the canon finally made the killing blow. Though my entire strategy was centered on avoiding the vamp lord so I also kept on moving and did everything I could to not give it a chance at combat. Harmonic convergence has also been a life saver.
7. Priests and A Lectors vs. Generals and Caps - The senior staff do much better in close combat as they can attack more with better stats and make units harder to brake (Hold the line), but the sigmarites make the unit they are in more durable and more combat capable (Hatred and prayers). Remember this against vamp lords. We cannot go toe to toe against a vamp lord so using a combo of these two to keep units in battle and able to kill more than they should is the key. I gave up on kitting the senior staff and the sigmarites for combat and rather give them durability. Ever since I have been doing better as the buffs keep the unit going.
Now for some tips on basic vamp things that you need to be aware of:
1. Do net let that bloodsucker get any spells of! You probably know this but I will say it anyway: Not only can he restore his lord's wounds, but he can bring entire regiments that you left for "dead" back to haunt you. It takes time, luck and good strategy to whittle down a unit of crypthorrors or graveguard and in the blink of an eye the damned magic can take all your effort and mix it with zombie manure and throw it in your face. Be especially vigilant for Van Hell's Danse Macabre. Moving an entire army 8" forward can screw the whole game for you.
2. Quite a few VC units have regen as their only defence, if you ignore their high toughness values, so use your prayers to get some flaming attacks against them.
3. They combat bleed, use this to your advantage

. Possibly due to desperation and resignation I have stopped trying to kill the vamp lord in close combat. He is just to powerful. So I always attack the weaker models in the unit he is with to get a higher combat resolution. Killing the vamp lord through combat bleeding. This is why most smarter VC players will never field a vamp lord in zombies and only more experienced ones will risk putting him in a skellie unit. It is just to risky. If demi's were to flank said zombie or skellie unit they would mince them down and eat the vamp lord for breakfast through combat bleeding. The VC's I play against use graveguard as their vamp lord's unit as they are much more durable and can do a lot of damage.
Overall strategy tips:
As danwhite said try and whittle his army down with shooting. If you are afraid of trying to snipe the vamp lord with the canon go for his other big nasties. Blackcoaches, varghulfs, crypthorrors and terrorgheists should be high on the priority list. Removing these threats can give you that edge to super flank and rear his general's unit and kill him through combat resolution and deal with thie smaller chaff in time. If you do try to outgun him do not move the first round of combat even if you go first unless you need to move some units to counter his deployment/movements. You will want to get the most shots off before he van hells to your doorstep. Do not try and use "elite" empire units, not that I think we really have any, against his elite units. Greatswords are horrible when compared to graveguard and demis were not meant to charge terrorgheists or get near it for that matter. Use a balanced list, a cavalry heavy army may thunder over elves and a massive artillery barrage my splatter orcs and goblins, but when it comes undead we have no such advantages as low toughness or low leadership. Yes their core is made up of zombies and skelletons, but their better units are not and you can only kill so much zombies with mortars before he starts raising the unit beyond its starting value.