In the novel The Adventures of Tom Sawyer, the main antagonist, Injun Joe, who is half Native American and half white, seeks revenge on the Widow Douglas because he is physically punished and humiliated in front of the town by her husband, Mr. Douglas, the town magistrate, when Mr. Douglas uses a riding crop to horsewhip Injun Joe for his crime of vagrancy.

These actions cause Injun Joe to become angry and vengeful. However, Injun Joe is unable to exact his revenge on Mr. Douglas because Mr. Douglas later dies, which is why Injun Joe then decides to target the Widow Douglas instead.

So, because Injun Joe can no longer seek revenge against Mr. Douglas, Injun Joe decides to take revenge on the Widow Douglas by plotting to mutilate her by cutting up her face.

Why does Injun Joe seek revenge on the widow in "The Adventures of Tom Sawyer"?

The answer to this question can be found late in chapter 29. Huck is on lookout duty when he sees two men sneaking about, and Huck decides that it is a good idea to follow the two men. Huck realizes that one of the men is Injun Joe. The two men stop outside of the Widow Douglas's house, and Huck is close enough to hear the two men talking. He hears the second man says that they should leave. It looks as if the widow has company over, so they can't carry out the plan. The plan is to somehow physically maim the Widow Douglas. Injun Joe doesn't want to kill her. That is not the right revenge.

Kill? Who said anything about killing? I would kill HIM if he was here; but not her. When you want to get revenge on a woman you don’t kill her — bosh! you go for her looks. You slit her nostrils — you notch her ears like a sow!

Injun Joe wants to exact revenge on the Widow Douglas because her husband is dead and no longer available for Injun Joe to exact his revenge on. Widow Douglas is the next best option. Injun Joe hates Mr. Douglas because Mr. Douglas had him horsewhipped in public.

But her husband was rough on me — many times he was rough on me — and mainly he was the justice of the peace that jugged me for a vagrant. And that ain’t all. It ain’t a millionth part of it! He had me HORSEWHIPPED! — horsewhipped in front of the jail, like a nigger! — with all the town looking on! HORSEWHIPPED! — do you understand? He took advantage of me and died. But I’ll take it out of HER.

The Widow Douglas had been married to Judge Douglas while he was still alive and active. As the judge in the town court, he had been involved with Injun Joe on numerous occasions over the years when Injun Joe had been arrested and brought before the court for one offense or another. He doesn't have any particular grievance with the Widow herself, but has great reason, in his perception, to be upset with the Judge. Injun Joe plans to take revenge on the widow because the Judge is no longer alive and able to pay for his insults to Joe's person and character.

He had me horsewhipped!-horsewhipped in front of the jail, like a nigger!-with all the town looking on! HORSEWHIPPED!-do you understand? He took advantage of me and died. But I'll take it out of her.