Well, for one thing, female Dunmer are know to be very.......promiscuous. That probably doesn't help you much though, huh?

On a more serious note, Dunmer are more magically inclined than, say, Nords, but not as fluent in the arcane as Altmer or Bretons. I'd say they're kind of a mix. Sort of a like a spellsword instead of a pure mage. That's not saying
all Dunmer are magic users, of course. Most of my Dunmer tend more toward the warrior side, because that's what I like to play. If you don't normally use magic, but want to for a change of pace, destruction might be a good school of focus.
Dunmer are also known for their ancestor worship. They respected their ancestors and the dead, so necromancy in Morrowind was somewhat frowned upon. Some Dunmer do practice necromancy, but never with dead Dunmer.
They also don't generally worship the Divines (Talos, Kynareth, Mara, etc.), instead focusing more on the "Good" Daedra: Azura, Mephala, and Boethiah. So maybe your character could do those quests while avoiding the "Bad" Daedra: Mehrunes Dagon, Malacath, Sheogorath, and Molag Bal.
They're generally received as being distrustful of other races, and can be vindictive. Before the loss of their homeland, they kept Argonian and Khajiit slaves and, after the Red Year, Argonians invaded Morrowind and took over most of the land. Perhaps your character has a thing against Argonians.
A quick skim-through of http://www.uesp.net/wiki/Lore:Dunmer might also help you out, too.
Just some ideas to maybe help get you started.
