Drop your jaw moderately. Touch the tongue tip behind the bottom front teeth and lift the mid-front part slightly toward the roof.

Americans pronounce eligible as EH-luh-juh-buhl (/ˈɛlədʒəbəl/). Stress falls on the first syllable — keep everything else short and quick.
Record yourself saying "eligible" and play it back. The mic stays on your device — nothing's uploaded.
4 syllables, 8 sounds. Tap a syllable to jump to its row, then explore each sound's mouth shape and how it's made.
Place the tip of your tongue against the alveolar ridge just behind your top front teeth, the same contact point as /t/, /d/, and /n/. The difference is what happens to the air: for /l/, you let it flow continuously around the <em>sides</em> of the tongue (that's why /l/ is called a lateral). Turn your voice on the whole time. Lips stay relaxed, no rounding or flaring. For the Dark L variant at the end of a syllable, also pull the back of the tongue up and back toward the soft palate.

Relax your lips, jaw, and tongue completely. Drop your jaw slightly and keep the tongue neutral.
The textbook way isn't wrong — it's just not how anyone actually says it.
Stress falls on the first syllable, not the others. Stretch EH — 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.