The first is renting a bicycle in Benjakita Park right next to the Queen Siriket Convention Center between Sukhumvit and Rama IV roads on Ratchadaphisek Road. If you are going there by Taxi be sure to either be dropped off at the Convention Center end of the park as that is where the bike rental stand is. The bikes are only 40 Baht an hour with a 40 Baht refundable deposit given at the time of rental. Also be sure not to forget some kind of ID Card as you must leave that at the booth when you rent your bike. There is a nice little bike path (Separate from the walking path) that goes around the little lake for you to ride on.
The second choice is ABC, Amazing Bangkok Cyclist. ABC is a bicycle tour company that tours Bangkok on bicycles. Although I have never had a chance to do this I have seen the groups of people riding around Bangkok and they always look like they are enjoying themselves immensely. They are located just off of Sukhumvit Soi 26, there is more information here.
Bicycle Rental Benjakita Park: