Ahh you face a rare TK opponent, one who knows how to use his army properly. Most people just think TK are underpowered and should be powered up. Imagine that?? I salivate at the thought, as my TK army went undefeated throughout its career.
How do you beat it? You need a LOT of magic defence. Magic really is the key here, if TK magic is left unchecked then they can wreck you by turn 2 or 3. Yes turn 2, I have had an opponent all but wiped out at the end of turn 2 before. Just as he must plan out his magic phase and decide which 2-3 spells he really really wants to get through, you must think about what he can do and decide what 2-3 spells you really really don't want to get through. Chariots sitting 8 inches from your flank? You don't want them to move. Skeletons sitting 10" away? Who cares if they move, let it go. Healing incantation to save a few skellies? Well I am sure there are more important things to stop. You should also never worry about the magic missile, it is weak and most TK players will only use it to draw out dispel dice.
Obviously another big factor in controlling the magic phase is killing one or two of the priests. You might definitely consider a suicide unit or two. A scar vet on cold one has decent speed, is immune to their fear and should be able to hit hard enough to kill a priest when he charges. Or use terradons if there are any priests on their own, which there often are especially if there is a casket.
Next and again very important, the heirophant. Like their blood sucking cousins, TK have a weak point in the army which makes them crumble. Unlike the VC, this guy is a really soft and squishy priest not a tough and crazy in combat vampire lord. He is worth destroying at all costs, literally. Not only will that stamp out a bunch of their magic, the army will start to crumble. Unfortunately, once he is dead you cannot breath a sigh of relief because they still have a general around to provide high leadership for the crumble test and can still pull off a win. But it is a big step, especially if you can get him early.
An engine will be invaluable here if you can hide it from the catapults until terradons/char skinks can deal with them. The burning alignment does extra damage against undead and will really really cause him some hurt. Early on the ward save will also be very helpful. Of course you need to hide it behind woods if you can to avoid it being shot at and deal with the catapults quickly. It is fine if it is still behind your deployment line in turn 3, but alive, you can then throw it out as quickly as possible and get close for burning alignment.
How to destroy the chariots? Well if you are killing priests and slowing the magic they should be a lot less deadly already. Sure they are still fairly fast, but nowhere near as good without magic throwing them into combat. Remember they are light chariots, a unit of them shouldn't be able to break through a rock of saurus in one turn, especially with cold blooded, your high toughness and decent armour. S7 kills chariots instantly, so great weapons on your characters will make them able to completely munch chariots. Stopping them getting the charge is also important, try to bait them with skinks to get them close enough for you to charge them.
I think the undead rule can be somewhat of a disadvantage to them, remember they can never choose to flee so if you manage to bait the chariots into a position where you can charge them, there is nothing they can do about it except sit there and die.
Finally, the princes and kings are flammable. Point a salamander at that chariot unit with the king if you can, should give him some trouble or at least make him scared.
Good luck, and I hope that helped.