I couldn’t find something about this in the players handbook (it is entirely possible I might have missed something), but is a character considered (within 5 feet) of another character if it is diagonal to it?
Given the wording here, I'm assuming you're using the, technically variant, rules for playing on a grid.
The basic rule for Space, found on page 191 of the Player's Handbook, says (emphasis mine):
A creature’s space also reflects the area it needs to fight effectively. For that reason, there’s a limit to the number of creatures that can surround another creature in combat. Assuming Medium combatants, eight creatures can fit in a 5-foot radius around another one.
When playing on a grid, 8 enemies surrounding a single person is easily represented by a 3x3 square, and would include the 'diagonal' spots.