In Much Ado About Nothing, why does Don John sabotage Claudio and Hero's wedding?

There are a couple of reasons why Don John tries to sabotage Hero's wedding. One reason is that Don John blames Claudio for preventing him from overthrowing his brother. Apparently, when Don John tried to overturn his brother's rule, Claudio, as Don Pedro's right-hand man, devised some clever tactics to prevent the overthrow and win this civil war for Don Pedro. Hence, Don John would love to get even with Claudio by hurting him in any way he can. The second and main reason why he wants to sabotage Claudio's wedding is that since Claudio and Don Pedro are close, by hurting Claudio, Don John will also hurt Don Pedro.

We see Don John blame Claudio for thwarting his attempt to overthrow his brother after Borachio tells him of Claudio's plans to marry Hero. We see Don John exclaim:

That young start-up hath all the glory of my overthrow. If I can cross him any way, I bless myself every way. (I.iii.54-56)

The phrase "young start-up" refers to the fact that Claudio has just started his military career, not only that, he has started as Don Pedro's right-hand man and gained a great deal of honor. When Don John says that Claudio "hath all the glory of [his] overthrow," he is referring to the fact that Don John has just tried to overthrow the prince's power and some how Claudio thwarted his plans, stripping Don John of the glory of being prince. Hence, we see that one reason why Don Pedro sabotages Claudio's wedding is that he wants to avenge himself on Claudio.

We also see references that show us just how much Don John resents his brother and would like to get even with him. Also in this scene, Don John complains about how he feels imprisoned by his brother, as we see in his lines:

I am trusted with a muzzle and enfranchised with a clog; therefore I have decreed not to sing in my cage. If I had my mouth, I would bite; if I had my liberty, I would do my liking. (25-28)

The phrase "trusted with a muzzle" depicts a dog being muzzled by its master to prevent it from biting or barking and portrays Don John as a muzzled dog. The word "enfranchise" means to grant liberty to, like a slave (Random House Dictionary). The word "clog" refers to a heavy block of wood that is used in lieu of chains (Random House Dictionary). Hence, the phrase "enfranchised with a clog" means that he has been freed like a slave who is still being burdened with a clog in lieu of chains. In other words, this passage serves to show us that Don John feels imprisoned by Don Pedro and that he is ready to strike at his brother again, which shows us just how much he resents his brother. Since hurting Claudio would also hurt Don Pedro, we can say that the main reason why he sabotaged the wedding is to hurt his brother.

How and why did Don John spoil the wedding in Much Ado About Nothing?

Don John spoiled the wedding because he was bitter and wanted to make his brother look bad.  He was the illegitimate brother, you see. Don Pedro was the legitimate brother.  Don John wanted to make trouble for his guests.  He had nothing against Claudio and Hero personally.  Hero's father Leonato was the governor, so ruining the wedding made Don Pedro look very, very bad.

Don John is a bitter and angry person.  He embraces this, and decides to take revenge on his brother and be the bad guy.  He seems to revel in it.

DON JOHN

...

I cannot hide
what I am: I must be sad when I have cause and smile
at no man's jests, eat when I have stomach and wait
for no man's leisure, sleep when I am drowsy and
tend on no man's business, laugh when I am merry and
claw no man in his humour. (Act 1, Scene 3)

Don John, as the illegitimate son, is not happy.  He seeks ways to make other people unhappy.  If he can make his brother unhappy, he will be happier.

Don John's plan was cold and calculating.  He arranged for Claudio to see Margaret having sexual relations with Borachio and told him it was Hero and another man.  Claudio flipped. He believed everything and renounced Hero on their wedding day.

The saddest thing about this is that as bad as Don John's actions are, Claudio's are also horrid.  Claudio waits until the wedding, then refuses to marry Hero.  Leonato demands proof that his daughter was unfaithful.

LEONATO

What do you mean, my lord?

CLAUDIO

Not to be married,
Not to knit my soul to an approved wanton.

LEONATO

Dear my lord, if you, in your own proof,
Have vanquish'd the resistance of her youth,
And made defeat of her virginity,-- (Act 4, Scene 1)

The truth is, Claudio has no real proof, because there is none.  He did not see what he thought he saw.  His word is enough though, to ruin Hero's reputation.  Hero faints and they tell everyone she is dead.  Don John has ruined the wedding, and he and Don John have destroyed Hero's good name.