So, my problem is:
When I record my voice with my monitor on, I hear a noisy in the background (similar to a radio).
When I record with my monitor off, I can hear my voice clearly, without noises.
I would like to record without these noises, but with the monitor on. If only I knew how to solve this...

Have you tried recording from further away from the monitor?

Move you microphone away from the monitor and make sure whatever cord you have connecting the mic is not tangled up with your monitor cord.

Does your cord go near the monitor? Is there any way to distance it/yourself from your monitor a bit?

EDIT: I don't know what you guys want from me, I'm PhoenixWright, not a double-ninja.

Change your sound-card, very often on-board sound card have very sub-par quality inputs/outputs with tons of parasites (at least the analogical ones).

To limit a little the issue you can try using a pre-amplified mic and/or disable your sound card microphone input amplifier if there is one.

I tried, but unfortunately this haven't solved the problem. I can still hear the noise. Yes. My sound-card is not on-board. And I'm a little picky about this, if I'll record my voice I want to hear it without this annoying noise. I don't want to minimize, I want to eliminate it completely.

I know that the problem is not my microphone, because I've tested it in other computers, and it worked perfectly. Maybe I should try changing some configurations in my monitor.

Can you direct you microphone (essentially put a load of foam around the direction you're not using, thus creating a directional mic). It sounds like it's recording some noise from the monitor (I think we all got that), so wrap a jiffy bag (or better foam) around it.

Probably not but . . . does your monitor have on-board speakers? Possibly disable or disconnect the feed line while recording if it does. There may be a way to turn them completely off on the set up screen as well.

I would also try recording with a different monitor hooked up just to confirm it is the monitor itself that is actually causing the problem rather than the cabling. Move any electronics further away, unplugging them might also narrow down the cause. . . =)