Although Henry V made English the official language of government, there is some debate as to whether it was him or his father, Henry IV, who was the first king to use English as a first language. On balance it was probably Henry IV (for the reasons stated below) so it follows that his predecessor Richard II was the last king whose first language was French.
The History of English article (from thehistoryofenglish.com) says ‘Henry IV, who came to the English throne in 1399, was the first monarch since before the Conquest to have English as his mother tongue’. The book The French in London says ‘Henry IV (1399-1413), the first king of England since the conquest whose mother tongue was English’(quoted by Jacquie Heys).
Henry IV would most probably have been influenced by his father John of Gaunt who was a patron of the English language. Also, by the time Henry IV was born, French was already being supplanted by English as the first language of the nobility. Douglas Kibbee says ‘French as a native language is definitely on the decline, even among nobility of Norman origin’ by the early 13th century (quoted by Jacquie Heys).
Considering the question though (last English king whose first language was French), we should also consider whether any kings after Henry IV had French as their first language. If we accept that Henry IV used English as his first language, it is highly likely that Henry V did too (given the general trend towards English).
Henry VI may be seen as a possibility because his father died when he was six months old (and therefore had no influence) and his mother was French (Catherine of Valois) but she did not have much part in his upbringing as she was not trusted by English nobles. Also, Henry VI’s father Henry V had made English the official language of government and English continued to be promoted during Henry VI’s reign (according to Douglas Kibbee). Then we should also consider that in the Hundred Years War, France was the enemy. The French in London says ‘Under Henry V, the interminable quarrel with France brought about a popular rejection of all things French’. None of this is conclusive evidence, but it is pretty strong. Under the circumstances, it also seems highly unlikely that the later kings Edward IV, Edward V and Richard III would have been brought up with French as their first language.
English was probably spoken by kings since at least Edward I who learnt it from his tutors (and his father Henry III also spoke it well). By the time of Edward III, it seems to have been widely used among the nobles, some of whom actually had to learn French from tutors.