I am trying to code a game in x86 Assembly. I don't know how to draw to screen using x86 assembly.
Based on my previous experience with assembly I assume I have to code a bootloader which jumps to a label that triggers a bios interrupt to draw to the screen.
I don't know what bios interrupt to trigger and how.
If you want "raw" (no operating system, not even DOS) access to the hardware, like the graphics card, you're in the realm of device-driver programming and device-specific code. For a "pure" learning experience - and something that is not unlikely to work from bootloader code for even recent PC-style hardware - you can try programming the graphics "as if it were VGA".
A good reference for this is the OSDev wiki, https://wiki.osdev.org/VGA_Hardware for VGA specifically.