How do you construct a 4 by 4 mini crossword?

If you mean a Word square, it's largely trial and error. Avoid letters that are hard to combine, like q and z. Pick a simple Word, like

STEP

Write another Word Word under it. It is often useful to alternate consonants and vowels.

STEP

LIME

This will start to suggest other Words to complete the square.

STEP

LIME

AMMO

WEAN

Put numbers at the starts of Words. 1-4 across, 1-4 Down. Write definitions. Bask in the glory.