First of all welcome to warhemmr.
Second of all you have really nice collection especially for starting player
Third of all I suggest that you make battle report (hopefully with pictures and with battle chronicler so we could help you even better)
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Character consideration Well now to the actual question. I make assumption that you are using the latest FAQ with 50% lord / hero allowance.
Eventhough his core units are really small and can be destroyed pretty much by anything. But as you probably noticed your opponent uses master of the dead to gain more troopers around and this is bad thing as we tend to win combats, but we tend to lose in attrition fights. Best way to counter this would use Level 4 Archmage with book of hoeth (and it would make your magic really dominant) Naturally if you are playing with 25% lord allowance there isn't enough points go for it. Lore of Light is always niche against undead, you can easily burn down the necromancers bunker. Lore of death is also really good since it allows you to snipe his general and something that Vampire Counts dread is the lose of general in early turns as they have relatively low LD (crumble will hurt like hell) and they lose the ability to march (well vampiric models can still march allthough his list doesn't include single one)
Another really good choice as lord (given that you have 50% allowance) is Loremaster of Hoeth with book of hoeth and some defensive gear. He will have ultimate toolbox of solving tons of problems (all of the BRB 8th signatures) in low level games and is decent close combat character.
All in all (50 % or 25 % allowance) I suggest to have only 1 character in this point values as our unit choices are relatively good without any character support and every elf body counts when fighting tons of undead.
Core considerations For core I would choose against the list shown either block of Archers (banner/musician) or Spearmen (full command) and fill it to the core tax value (so 23 archers or 25 spearmen). Reason is simple the grinding of weak undead core. With archers I would try to concentrate every fire to the necromancers bunker (with support of magic tries to get rid of it) with spearmen just went to fight with the skeletons etc. You are doing decent ammount of damage to them with higher WS rerolls and extra ranks of attacks and have rank bonus. Something like silverhelms just lacks push to do this.
Secondly if you are having your General in the spearman block (especially if the general would be Loremaster) banner of discipline is good investment with +1 LD turning your generals inspiring precence to 10 (if having lord choices) is really good, since you are taking tons of LD test (due fear) and it is highly unadvisable to take BSB in such low games that +1 LD can really save your day.
I love reavers for redirection duties allthough in so small games as this they give imo too much points as mere sacrificial unit. Allthough even 5 of them in flanks could help you grind those core units for extra combat resolution.
Special considerationsI think against the list shown White Lions would be the best choice to tackle either Grave Guards or Black Knights. With Str 6 attacks before the enemy strikes (well vampire strikes first due higher initiative) they can lay waste.
Swordmasters are pretty much equal to white lions in this fight. Swordmasters would be better when fighting their core, due higher number of attacks. Especially if he would manage to get flank charge with some ranked core swordmasters have capacity to strike wounds to flanks.
You can add some simple magical weapon to the champion incase that cairnwraith cannot be dealt with magic missiles or other magical attacks.
Rare You don't have much rare choices to go with, but in general Eagles could be good choice in this game due low point cost and easily redirecting.
I am not fan of using Bolt Throwers, but placing them in the far side to force your opponent to move the black knights so that their flank isn't showing up is really good investment and with the ability to shoot 6 normal str 4 strikes, it can be used to kill grave guards.
Frosthearth phoenix would be something that your opponent would be really, really frustrated. Locking the vampire / black guard unit in combat with it or using it to help any other unit to kill grave guards. He doesn't have really any good choices to fight against.
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Well my list would be (adjusted to your models

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Lord (50% allowance)
Loremaster of Hoeth , Book of Hoeth, Talisman of preservation
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core
25 spearmen full command banner of discipline <- Loremaster here
5 x spear reavers (hope of making direwulf vanguard little harder nad and killing the direwolfs on charge and then redirect either grave guard or black knight bus so your swordmaster can fight the other without interference.) If you get eagle I would probably choose it instead, but eagle cannot kill so big dire wolf unit., but eagle can be used for more important redirecting.
special
21 x swordmasters (7 wide), full command, banner of swiftness
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For deployment you don't have much choices as he will probably deploy his core units first, but try to get spearmen to the swordmasters side where black knight bus is. Spearmen can easily stop take a charge from the blackknight vampire buss and hold for couple turns. Reavers would be best to be in either flank and hopefully at the same side where dire wulfs and black knight bus is.
depending what kind of terrain you are going to use, but try to get as narrow cap as possible where your units manage to go side by side without too much fearing of flank charges. Think as Spartan king Leonidas and yell "This is Ulthuan!"

Hope this would help you

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