How to show a character in black and white mode with colorful background?
Have you already filmed your scene? If not, then the easiest way is to film your character against a blue or green screen, key out the background, apply a monochrome filter to the character, and insert your other background. 

If you have already filmed it, you can try using a color correction filter on the character, but it's very unlikely you'll be able to easily isolate the character properly with just a filter, unless the background is a solid color which doesn't appear anywhere on the character.

So your other option is rotoscoping which is very difficult and often tedious work to look good.

What software are you using, by the way?