There are 88 total screens; the Crash Site screen does not seem to count. To visit all of them, besides from visiting the usual screen, jump around and visit a few screens that do not have names. Note: Do not skip the Treemen's Domain.
The three heart containers are found at the indicated locations: (Screenshot 2) (Screenshot 3)
You must find a certain number of different screens. Pass through all the cores, artifacts and secret locations. Make sure to telejump out of screens when possible. Jump as far as possible to reach the furthest position of every screen to check for hidden locations.
Do not any checkpoints after dropping down to the Alien Outpost. Hit the checkpoint located here , collect the Gauss Gun, then intentionally die. Go up and reach the top right part of the secret cliff (Screenshot), then kill the Chomper to get the upgrade for the Gauss Gun (Screenshot).
Walk back and forth at the end of a screen to the next one repeatedly.
Enter the room with the treeman and two mirrors facing each other . Get on the small area between both mirrors and repeatedly fire teleshots.
You must travel from the start to the inside of the Alien City without activating any checkpoints. Set "Autosave" to off in the options menu. If you touch a checkpoint or die, quit without saving and restart from the beginning. Use shortcuts to bypass dangerous locations and checkpoints.
Every alien must be spoken to, including the Stone Alien, the Treemen, and the Alien Guards. However, Grigori does not count. Note: Some aliens may be difficult to reach.
You actually have to kill every single monster in every screen, except for ghosts, at least once. Note: Do not bypass all the monsters before the alien city.
It is not necessary to have the Gauss Gun upgrade. When Grigori gets low on health, hit him in quick succession. Shoot him one or two times then dodge his bullets. Then shoot him another one or two times for him to change sides. A total of three hits are required for Grigori to change sides. You must force him to change sides four times before being able to kill him.