How do you find chords to fit your melody in a particular genre you want to compose?

Start by figuring out what key your melody is in. From there, play the chord of the tonic. When it sounds like you should be playing something other than the tonic, look to what notes are going on in the melody for hints. It's good to know common patterns to help you make educated guesses.

You use your knowledge and skill to find chords that fit your melody and give the sound you want.

There really is nothing else to it.

You should take a class in music theory before trying to do that.  Every melody note could be a part of three major chords, then there are minor chords, diminished chords, and others.  There are a lot of things you could do, and rules are broken frequently, but the thing is, will people want to buy it? 

Just check out the I - IV - V - vi. (One, four, five, and six minor chord)

Those four make up a mind boggling number of hits over the past couple hundred years.

That’s the formula, anyway, but do a small bit of exploring on chords that relate to one another. Essentials of music theory.

For example, if I start on C major, I already know there are six other chords in that key I can pick. Two other majors, three minor, and one diminished.

 And the I IV V vi? C - F - G and Am

Those four chords always mesh. In any order.

A quick search for hits with my random chord picks?

“Complicated,” Avril Lavigne

“Whistle,” Flo Rida

“It Girl,” Jason Derulo

“Otherside,” RHCP

“Hurt,” Johnny Cash

And a zillion more 

The thing is music is like drawing in a lot of ways. You can just listen for what sounds good. Having them memorized just speeds that up.