Yes there's McClure, and I think there is another short one, towards a reservoir to the north - can't remember what it's called & haven't done it.

There's a hostel in Aspen.

There's also a hostel in Leadville, which would be a good starting point for Copper Triangle. THere's camping on Lake Dillon near silverthorne.

Personally I woudn't choose Aspen, it's too limiting. The 2 climbs are nice, but the town is outrageously expensive and the camping doesn't have many amenities. If you stay in the DIllon/Frisco area, there are more climbs (Loveland Pass, Hoosier Pass, COpper triangle) and a nice ride north on Hwy 9 towards Kremmeling (Ute Pass), and you can drop down to Idaho Springs and climb Mt Evans or Berthod Pass. And then (most importantly) there's the Butterhorn Bakery in Frisco.

Of course, you don't have to stay in one place - you can spend a couple days in Aspen, A couple days in Frisco, a couple days in the Front Range....