If you can think of Dr. Jekyll and Mr. Hyde as brothers, the significance of the Cain and Abel allusion becomes easier to see.
In Stevenson's story, Dr. Jekyll is Abel and Mr. Hyde is Cain because Jekyll is the good "brother" while Hyde is the bad "brother." In Stevenson's story, as in the Biblical account, one "brother" will kill the other. In this case, Dr. Jekyll will commit suicide because that is his only way to kill his evil "brother" Mr. Hyde.
Another point of similarity is that, in Genesis, Cain kills Abel out of jealousy while in Stevenson's story, Hyde wants to "kill" Jekyll by taking over his body and his life.
In a sense, then, Hyde tries to be Cain and kill Abel, but this time Abel will win (sort of) and kill Cain (though he himself dies as well).