Hotel lodging is available at Klamath Falls with plenty of choices for all budgets. I like the Olympic Best Western, it's clean, quiet, and has wireless connection.
Camping is available at Crater Lake (south entrance) with spacious sites. We had no problem finding a spot without reservation on a 4th of July weekend couple years ago. The site was beautiful, very quiet, and quite secluded from adjacent sites. There's a store on the campground.
I haven't biked there but you could use the campsite as a base if you are not scared about the climb to the rim. The rim is an up and down road with outstanding views. I would expect the road to be busy during summer time but it shouldn't be a problem if you start early in the morning. I was doing some photography at sunrise and was almost alone on the rim.
There are plenty of rural roads around the park, the area around the north part of klamath lake is pretty. South of Klamath roads tend to be long straightaways, so a bit boring for riding. There's a road going into Lavabeds Park that could be interesting but I have never driven it.