I was originally going to limit this to the two tetralogies, until I realized that the Richard II tree included Henry III, who appears in King John. Here’s a tree for the characters in that play. I’ve tried to make it a little more compact than my Richard II tree, since the plan is to put all these together into one big tree in the end. Also with that end in mind, I’ve marked the King John characters with a “J”; later all the characters will be marked according to the plays they appear in.
By the way, don’t confuse this with a real genealogy, since some of Shakespeare’s characters (the Faulconbridges, for example) are fictional. I’ve included one real-world fact that Shakespeare probably didn’t know, though. The mother of Henry II’s bastard son William Longsword, it has recently been discovered, was almost certainly the wife of Roger Bigod, Earl of Norfolk — who, in turn, is presumably the same as the “Robert Bigot” in the play, there having been no other Earl of Norfolk with a similar name during John’s reign.