You can use FL Studio with a Gameboy soundfont available from here. If you're going for authenticity, you'll need to keep the GameBoy limits in mind, especially with only allowing one note at a time and only having 2 square channels, a noise channel, and a wave table channel playing at once.
If you want genuine authenticity and aren't unfamiliar/afraid of trackers, you can try a GameBoy tracker. There's a list available here. Since they run on actual hardware (or an emulator), the sound would be 100% authentic. I only tried Fat Ass which is a ROM. I couldn't figure it out as I'm not very well versed in trackers outside of FamiTracker, but I managed to load an example song it provided and sure enough it was authentic.
You could also try this GameBoy tracker for PC, but I couldn't figure it out for the life of me, however I'm sure some dedication would fix that.
I would personally recommend just using a soundfont with FL Studio/DAW of your choice as the process is much simpler.
Last year I used FamiTracker (and I will use it again this year too).
Basically make a new file and change it to use n163 (3 or 5 channels). N163 has up to 8 channels of wavetable audio like the Game Boy's 3rd channel, so we can use one of these for that purpose. The other two or four channels would be used to mimic stereo.
In my previous FamiTracker file from last year,
FT Pulse1 = Game Boy Pulse1
FT Pulse2 = Game Boy Pulse2
FT Noise = Game Boy Noise
FT Namco1 = Game Boy Wave
FT Namco2 = Stereo (Left) - can be either Pulse 1 or wave
FT Namco3 = Stereo (Right) - can be either Pulse 1 or wave
To make there stereo sound work, you export to NSF format and play it in a NSF player that lets you change the stereo settings per channel, then simply move Namco 2 and 3 to Left and Right, respectively. You have to make sure that only 1 note per channel is playing though to make it sound authentic, so if I have something on the Namco2 (left stereo) channel, I need to mute either Pulse1, Pulse2, or Namco1 depending on what I'm having play on the left stereo.
Use this as a resource for the different effect types. I figured out how to use FamiTracker over a long time by just getting used to it, so not sure how to instruct someone how to use it in a short period of time. You could try Danooct1's famitracker tutorial but it doesn't really go into too much detail either.
It uses a piano roll instead of a tracker and has a few knobs to twist around, so it's more mouse-focused.
As for trackers, I recommend LSDJ on a Game Boy or emulator. It's just so much fun to use. Like playing a game but you also make music.
