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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?

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Yes.

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.

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    \$\begingroup\$ As a note, there are also different rules for diagonal distances on a grid. You can have it take 5 feet of movement for each tile moved diagonally, or have it alternate between costing 5 feet and 10 feet. The latter is more mathematically accurate than the former, but it can potentially bog down movement with a touch of math, whereas counting tiles with the basic of 5 feet in any direction is quicker. This can be relevant if creature have reach, as a 10 foot reach may allow you to reach a creature two tiles on front, or one diagonally and one in front, but not two diagonally. \$\endgroup\$
    – Iter
    Feb 7 at 19:49
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    \$\begingroup\$ @Iter This is true, and I almost included it in the answer before deciding it made things too cluttered, given the extra optional rules are for movement. I also have never played with a DM that used the technically-more-mathematically-correct optional diagonal rules, which at least gives me the impression that it's not particularly common. Very useful addendum for those who do, however. \$\endgroup\$ Feb 7 at 20:05

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