Two options: a guitar with MIDI output, or converting the guitar audio to MIDI
The first option is a guitar with a MIDI pickup. This is usually expensive and varies in performance. I have a Roland GK kit with a GI-20.. pretty old stuff, and it is not totally reliable. You have to play very deliberately and it does not do legato/bends reliably. You have to hit notes pretty hard to get them to come through reliably. Newer hardware is probably more reliable. I would be willing to bet that Aethernaut uses a violin with MIDI hardware built-in.
The second option is trying to use software to convert audio to MIDI. This also suffers from reliability issues, though honestly I haven't tried any software for this in at least 8 years.. it might have come a long way since then.
If you're going for something that will work in real-time, LSDJ probably isn't your best option as you'll have no control over dynamics... it won't matter if you play notes louder or quieter, they'll all come out of the GB at the same volume. mGB would probably be more reliable but lacks in some sound options.
Either way you'll need an Arduinoboy to get the MIDI into the Game Boy.