Raise the back of your tongue to touch the soft palate (velum). Stop the air, then release.

Americans pronounce compute as kuhm-PYOOT (/kəmˈpjut/). Stress falls on the second syllable — keep everything else short and quick. You'll hear it in sentences like "Use the computer to compute the future value".
Record yourself saying "compute" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 6 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Press your lips together to stop the air, then release. No vocal cord vibration.

Start with the tongue mid-front raised high, almost touching the roof of the mouth (but not touching). Glide into a tight lip circle as the tongue back lifts.
Touch the tip or front edge of your tongue to the roof of your mouth just behind your teeth. Keep your jaw relaxed. Stop the air, then release with a puff.

The textbook way isn't wrong — it's just not how anyone actually says it.
In "compute", the "t" is not released — the articulators get into position but hold without the burst of air. Air stops but there's no release burst — the articulators hold position.
Stress falls on the second syllable, not the others. Stretch PYOOT — keep everything else short and quick.
Don't pronounce the first syllable too fully. The unstressed syllable reduces to a schwa — the lazy "uh" sound — in casual speech.