1) I assume you're taking the headphone output of the GB and plugging it into the Mic input of the computer, and that's a big part of your problem. Headphone outputs are already amplified, and you're putting it in an input that has a cheap pre-amp on it. Use the Line input.
2) Put the GB's volume to the maximum, that will make the signal much louder than the noise.
3) If you want to nitpick, plug in the gameboy, volume to the max. Then monitor the input and apply a noise gate. Move the threshold up until it cuts the noise. Then test it out with music, and adjust it if it cuts off some of the music.
4) Use a noise reduction plugin. Record some noise, empty of music, and use that to make an image of your noise. Then you use that to filter out this noise print from a recording. Don't go too crazy with it or it will also remove a lot of the music.
5) Noise is part of any recording. You can fight it, but there will always be a bit of it. The GB has a crappy amp and crappy DAC. Same thing with the built-in soundcard of any computer. Crappy DACs. Your signal goes from Digital to Analog in the gameboy, then from Analog to Digital again in the soundcard. Crappy pieces at these points will always add noise. You can alleviate the problem by getting a decent soundcard with good DACs.
In your case most of the noise comes from the gameboy. And using the Mic instead of Line input doesn't help at all.
