There are plenty of characters that I'd like to play but have never rolled up, but I'd really like to play more warforged or dragonborn characters, or characters of classes I've never played, like a 4E Battlemind or 3.5 Druid.
So for this post, since it specifies one character I'd like to play, I'll go ahead and choose a human artificer of House Cannith in an Eberron campaign. Much as I love Eberron, I haven't gotten to play in that setting very much, and House Cannith is one of my favorite setting elements. So let's see, a human artificer of House Cannith, and if I had to set some sort of goal for that character, it would be to go on an expedition to Xen'Drik to find information on the ancient warforged of the old giant civilizations.
I know I could write up a longer story for this character, but that's plenty to get started. If I ever do make a Cannith artificer, I'll tailor it to the campaign rather than having a pre-written character already worked up.