Unicode contains sections which specify emoji as "characters". They're regular characters, you only need a font which can display them. Also see the Unicode Emoji FAQ.
In a text file, characters are basically encoded as numbers in the form of bytes. To display those visually on a computer screen you need a font which contains the visual glyph to render this character. Since the process is always numeric identifier → font → visible glyph, it should be pretty obvious that a "character" can be anything visual, including emoji or any other image.