There are two answers to this question, in my opinion. Firstly, if we examine the very revealing scene of Act I scene 3, we discover more about Don John's character, and realise, as he himself says, that he is a "plain-dealing villain." Therefore, in a sense, he is a man who would look to cause problems and make mischief with anybody as he is someone that is committed to spreading evil and discord amongst those who he is with, and especially his brother, who, let us not forget, he has just tried to unsuccessfully rebel against.
Secondly, however, in the same scene, we see that one of the reasons why Don John is so delighted that he can launch a stratagem against Claudio is because Claudio has gained much through Don John's fall in favour. Note what Don John says:
That young start-up hath all the glory of my overthrow. If I can cross him any way, I bless myself every way.
Therefore there is a sense in which getting back at Claudio will satisfy Don John's desire for personal vengeance.