KIll their mages at the earliest opportunity! Use chaff to hold things up. Roast everything with Sallies. Never give them a fair fight. Go two on one against their units, using Steg or Scar Vet to push you over the top on combat res. I have found Skrox units work well, staying steadfast, sacrificing skinks while the Krox str 6 hits cost him dearly in higher cost troops.
There are really only 2 units in the HE army that are a threat: white lions (WL) & swordmasters (SM). Both units can beat lizzie infantry in a straight up fight. Since WL are highly resistant to normal shooting, they are the priority targets for sallies. In addition, they are stubborn and thus cannot be broken easily and must be ground down to the last elf most of the time. About the only effective way I have found of fighting them in hand to hand is to hit both flanks of the WL unit (not easily done) to prevent a reform and limit the number of attacks back at your own units. A steggie just on the corner in contact with a single WL will speed this along nicely. The SMs have much less protection and massed skink shooting will hurt them badly, especially if you can touch a horde with a sallie template or two prior to shooting. Also, they are not stubborn and thus can be broken and run down if engaged tactically (i.e. buffed saurus/TG unit and/or flank charge). Magic - HE have the most potent caster in the game in Teclis since he can cast with IF on any successful casting that includes a double along with a number of other bonuses. Teclis can single handedly ruin entire units with dwellers/purple sun/okham's mindrazor, etc. On top of that, at the point levels you are talking about they are likely to bring the BSB with a banner that makes the unit immune to all spells, thus making it almost impossible to kill Teclis since he will be bunkered behind a barrier of WL units. Another common theme is a "Coven of Light" list that include 4 wizards and the ability to take multiple banishment spells. This allows HE to pump out some very powerful magic missiles (S7) along with the default MM spell. As has already been noted by others, the key is to close with and destroy the wizards even if it means sacrificing a unit to do so. Overall, HE can be a handful but they do die like flies once they have to endure any sustained ranged attack so pile on the poison and sallies.
Salamanders are good, but watch out for Dragon Princes. They are nigh fireproof and very fast. The handsome forumite, Scalenex, has made a Tactica with unit by unit breakdowns of the High Elves found here: http://www.lustria-online.com/threads/high-elf-tactica-detailed.9112/
Try casting Dwellers on his lions, and use Sallies on his sword masters. You should whittle them down before engaging and Flanking them with buffed Saurus.
thats unwise. the white lions are more resistant to dwellers by far (S4). they will only get a 6+ save against salamanders, so if you do the other way around (sallies on lions and dwellers on swordmasters) you will kill more models.