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.