Press your lips together, add vocal cord vibration, then release.

Americans pronounce brilliant as BRIHL-yuhnt (/ˈbrɪljənt/). The L in "brilliant" is a dark L — the back of the tongue rises toward the soft palate, adding a small "uh" quality before the L. This is called the Dark L vs Light L, and it's one of the defining features of casual American English. It comes out as BRIHL·yuhnt. Stress falls on the first syllable — keep everything else short and quick. You'll hear it in sentences like "Her lawyer presented a brilliant argument".
Record yourself saying "brilliant" and play it back. The mic stays on your device — nothing's uploaded.
2 syllables, 8 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, add vocal cord vibration, then release.

Curl or bunch your tongue without letting the tip touch the roof of your mouth. Brace the sides of your tongue against your upper back teeth, and round your lips slightly.
Drop your jaw slightly with relaxed lips. Touch the tongue tip behind the bottom front teeth and arch the top-front toward the roof.

Keep the tongue tip down and pull the back of the tongue up toward the throat. The 'dark' sound comes from the back.

Lift the middle of your tongue toward the roof of your mouth, but stop just short of touching. /j/ is an approximant, not a stop. The tongue tip stays down, lightly resting near the back of your bottom front teeth. Voice runs through the whole gesture, and the tongue glides smoothly down into the next vowel. The lips stay neutral or pre-shape for the upcoming vowel (rounding early for OO in <em>youth</em>, for example).

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
The schwa before N disappears — N becomes the vowel of the syllable. Go straight from the previous consonant to N.

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.
The L in "brilliant" is a dark L — the back of the tongue rises toward the soft palate, adding a small "uh" quality before the L. Dark L adds a small schwa-like "uh" before the L. The back of the tongue lifts toward the soft palate.
In "brilliant", the short unstressed vowel before "n" disappears — the schwa is absorbed and the "n" becomes the syllable nucleus on its own. Schwa is absorbed — consonant becomes the syllable nucleus.
Stress falls on the first syllable, not the others. Stretch BRIHL — 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.