CDs are my second army. Some general tips: don't buy flaming attacks, buy flaming protection, don't take monsters, field WM hunters (flyers, chamos) and Tetto, you can forget all our rare options, take mobile units if you can.
CDs are quite versatile: they can field a gunline, good infantry blocks, hordes, powerful monsters, they have good magic phases BUT they are very expensive, so they have to choose what to do. Facing a gunline you'll probably notice that they cannot protect their WMs properly unless they field hobgoblins: dwarfs are pricey. You should have the possibility to hunt his WMs, just don't engage hellcannons with flyers or chamos. A CDs gunline will destroy monsters with ease, and will obliterate most infantry blocks, so be careful: CDs WMs are brutal and accurate.
CDs are slow... maybe. Yeah, dwarves are slow (m3), but hobgoblins have m4, bullcentaurs have m7, iron daemons have m6 and strange movement rules, so be careful. Oh, destroyers and wolf ridefrs have m9. Usually, though, CDs are slow because both their core choices are quite slow (dwarfs and hobgoblins), so it's unusual to face running lists, just don't think your enemy will not have fast and heavy hitting units and chaff, because he could field them.
Skinks shine versus CDs, while saurus warriors and TGs struggle. CD players fear and hate our skinks and our flyers: probably lizardmen and woodelves are their best counters.
Destroyers hit hard, and killing them could be difficult. You coukl redirect them, but if you want to kill them you can try with a mounted 2++ vs flaming character, skink fire or thoughness lowering. Don't let this guy hit your units unless we're talking about chaff
Care for iron daemons, they can lock your main unit in combat for the whole game. They're not as good as steam tanks in combat, but they're very though and have a tricky movement phase. You can try to destroy it (metal magic, kroxigors, a mounted character with high strenght) or to lock him before he locks you
LM characters are better than CD ones, regarding both magic and combat, but remember wizards are engineers too and can fight quite good if properly equipped. No fighting lords, but CDs fighting heroes can be quite though or annoying
