I think it's either very doable if you're flexible, and tough if you're very picky. There are plenty of apartments no matter what time it is, and I would imagine that you are competing with (Brown) students mainly if you're looking for the College Hill, Fox Point, or Wayland areas. PC is further Northwest I think and thus not so much in the "high demand" areas, and students from the other schools (RISD, Johnson & Wales) are also more spread out.
So, I would say it depends on where you're willing to live, what you're willing to spend, and what quality you're looking for. In a week, you should have no trouble finding a decent but generally older 1 or 2 bed in the $1000+ range in the top areas, or a nice place in a sub-$1000 range in an outskirt area (E. Prov., N. prov., the northern reaches of East Side, Pawtucket, etc.).
As far as the bike, I don't see why you couldn't keep it onsite ... most of the apts I saw came with parking. Still, many do not, so always ask.
It also depends on how much time you have in that week, of course. If the whole week is devoted to this, you shouldn't have trouble finding a place even if you're picky. You could also say it depends on the housing market, but that's a big guess/gamble. Some say that the condo market will dip significantly towards fall ... in which case maybe there will be more people looking at buying, or more condo buyers will have condos ready to rent.
Lastly, depending on what kind of urban life you're looking for, I would suggest checking out the west side of Fox Point (kind of a small vacation town feel mixed with a liberal college feel, with lots of little shops and restaurants) or the Angell St. area near Wayland (close to shops in Wayland Square and Thayer St). Both locations have that nice walkable feel, but are also close to downtown. If you're looking for more urban than that, Federal Hill is a denser area with lots of Italian influence, and is also close to downtown. Of course, if you're looking for *city*, you could live in the Downtown area as well ... but with bikes, I doubt that's what you're looking for.
pix