On the subject of claims to the throne, it has to be said that in the late 14th century the rules had not been fully worked out. It was fairly clear that a king would normally be succeeded by his eldest son, but what would happen in the event of a king dying without a son was far less clear.
It appears that towards the end of his reign Edward III purported to entail the crown on John of Gaunt in the event of Richard II dying without heirs. Rather illogical, since Edward had been loudly laying claim to France since 1340 on the basis of inheritance through his mother. However, logic is not always a strength of the world of politics, and the reality was that the king was senile and under the thumb of John of Gaunt, so that may have had something to do with it.
Richard II apparently decided that his successor should be Roger Mortimer Earl of March, who was the grandson of Lionel of Clarence, Gaunt's elder brother. This is faithfully recorded by the Westminster Chronicle, but the issue does not appear to have been entirely settled, and one suspects that Gaunt, his son Henry Bolingbroke, and maybe Thomas of Woodstock, Duke of Gloucester, the king's youngest uncle, had other ideas.
Towards the end of his reign the question became even more open. Gaunt was dead, Bolingbroke was banished and declared a traitor, and Roger Mortimer, killed in Ireland in 1398, had fallen from favour and been recalled before his death, probably to face the music. Though Roger had a son (the same Earl of March involved in the Southampton Conspiracy) it seems Richard at this point made Edmund of Langley his heir. This is certainly the belief of Ian Mortimer in Fears of Henry IV and if he is correct it the House of York opened the Fetterlock rather more completely than I thought!
More on Mortimer's book, and Henry's rather dodgy claim, at another time. Suffice it to say that Henry IV entailed the succession on his heirs by parliamentary statute, the first sovereign to do so. (The practice later became quite fashionable!) When Edward IV succeeded, however, he did not enact a succession statute, because he believed he was the legitimate heir of Richard II, through his Mortimer grandmother. By modern succession arrangements, at least, he was correct.