Basically you need to kill the boss that looks like a "big dumb bird".
I recommend the grenade launcher for this but I don't have all the weapons yet so other weapons may be better.
To reach this specific boss, you need to choose the path that has a flying bird monster (not the one eyed one) next to it at the beginning of the run.
Once you reach the boss, don't exit the snake (the tube thing that teleports you between levels) to stay invincible. Press once
to stick your head out and shoot at the boss (target head or body). If your ammo is about to run out, wait until it's clear, then quickly reach the flowers for ammo and head after to the top left corner while shooting at the boss. If it is not dead yet, jump over it to the other corner and keep shooting till it's dead.
I will update this guide frequently.