Here are the things you are likely to face against when playing against skaven: doomwheels, cannons, hell pit abominations, a warlord with the felblade in a bunch of clan rats, and a level 4 with a bunch of storm vermin. And probably 2 buses of slaves (or so).
For doomwheels and aboms: They are weak to skirmishers. two groups of skirmishers can railroad them off the board pretty easily while whittling them down with poison shots. To do this, you need to surround them and place yourself as close as you can on the side, but out of their charge arc. They can't pivot toward you to charge you. This is actually the best time for blowpipes, since it gives you the extra shots and you can't stand and shoot against them anyway. The doomwheel's 3 shots will take out 3 skirmishers, but that's only 21 points killed off a turn (not a big deal). Just don't let them charge anything expensive, because they have a very good chance of wiping out expensive stuff.
For the cannons, don't bring an ethereal slann. The cannons are magic attacks and you don't want your general's survival to be dependent on a coin flip. Maybe bring a nice big skink cohort for him to hide in. Temple guard are nice (as always), but they are a juicy target.
Bring group wiping spells. Dwellers, Pit, and Fiery Convocation are all your friends. He's going to have a ton of stuff in just a couple of units, so these really get their money's worth.
Don't throw scar-vets against doomwheels or the felblade. Scar-vets DO NOT LIKE these guys.
Try to avoid getting tarpitted by the slaves if possible. It takes FOREVER to kill enough of these guys to get them to explode (they don't break, they just explode).
Salamanders are hilarious vs. skaven (unless you are the skaven player).
Expect a lot of his stuff to either wipe out a lot of your stuff, or wipe out a lot of his stuff. I never seem to beat skaven. They always seem to beat themselves.
A unit of terradons or rippers work well as cannon hunters. Bring two, keep them far apart. If the cannons try to take them out, that is one turn they aren't shooting at important things.
Prioritize dispelling skitterleap, especially early on. He may drop a character right next to your slann/TG unit (if you take one) and throw a brass orb, which is basically a thrown Purple Sun.
Finally, DON"T BRING YOUR TG UNIT CLOSE TO HIS GREY SEER. Dreaded 13th is a close range spell, and it will destroy you. You don't want to lose your slann to this, and if he wipes you out, he gets a brand new unit of clan rats.
Skaven are fun to play against. They have very clear strengths and weaknesses, and pretty much everything has a nasty miscast chance to make up for their absurd strengths. Hopefully this helps you out.