Both are often named as either sons of either Poseidon or of Glaucus the son of Sisyphus. Chrysaor is always known as the son of Medusa and twin brother of Pegasus, the same Pegasus who was tamed by Bellerophon. Both have very little stories in mythology, though they may have had more that have since been lost to time.

Bellerophon is known for killing his brother (or friend or enemy or something, there's a bunch of different versions) and had to atone for his sins by serving a king (much like Heracles). The king's wife lusted after Bellerophon and claimed he tried to force himself on her after he refused her advances. The king sent Bellerophon to another king to kill him (as he didn't want to anger the gods by killing a guest), so he sent him to his other king-friend who also refused to kill him for the same reason. The other king then sent Bellerophon to kill the Chimera (believing he would die of course), but Bellerophon slept in a temple of Athena where he had a dream of Athena giving him a golden bridle and he awoke with it. Bellerophon also had a run-in with the Amazons and came out victorious. Before facing the Chimera Bellerophon tamed the Immortal winged horse Pegasus (the brother of Chrysaor), and the two faced the Chimera with Bellerophon killing the beast with his spear (or in some versions had a piece of lead at the end of his spear, that was burned over the mouth of the Chimera and it suffocated and died). Bellerophon then flew Pegasus up to Olympus, only to be struck down by Zeus. Pegasus got to stay on Olympus though.

Chrysaor was born alongside Pegasus, from the neck of Medusa, after Perseus decapitated her. Chrysaor is seen as either a giant, a demigod in golden armor with a golden sword or a gigantic winged boar with golden tusks (or in one version I read he was one of the younger cyclops children of Poseidon, but I can't seem to find that one anywhere). Chrysaor is the father of the three-bodied giant Geryon and a follower favored by Ares, who often accompanied him into battle. And that's all there really is on them.