Replace the TV, it it only has RCA connections it will be a horrible picture using a Bluray player. You can find a good used 32-40" HD TV for under about $200.
If your "RCA" connections are actually Component connections which can do HD, and the cable box has an HDMI INPUT (not just output), you can just route the player though the cable box.
If you don't have Component connections on the TV, replace the TV if you want to use it with a Bluray player.