Don John, having unsuccessfully rebelled against his brother, Don Pedro, and been defeated in battle, turns to other stratagems to make trouble. He feels especially malevolent towards the "exquisite" Claudio, who has risen in Don Pedro's favor since Don John has fallen. He resents Claudio's power and privilege and wants to bring him down simply for the pleasure of making him miserable. This is called spite.

Don John, speaking to his henchmen in act 1, scene 3, offers clear insights into his own character. He feels angry at Don Pedro, is not a gracious loser, and says he would rather be a cancer "canker" to his brother than a rose, an asset. In other words, he would rather spoil happiness than create it; that is the kind of person he is. Don John also says he would rather "disdained" by everyone than to pretend to be lovable. He doesn't want to wage a popularity contest against Claudio by pretending to be nicer than he is. He simply wants to be his angry, vengeful self and make trouble for his brother's favorite:

I had rather be a canker in a hedge than a rose in his grace,

and it better fits my blood to be disdained of all than to

fashion a carriage to rob love from any.

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.