觉 (traditional 覺) caps 见 (to see) with the studying-roof from 学 (to learn): seeing that has been learned — awakening, awareness. From 'waking' it split two ways: the sleep you wake from (jiào) and the feelings you wake to (jué).
One character, two lives: read jiào it's sleep — 睡觉 (to sleep); read jué it's feeling — 觉得 (to feel/think, the phrase 我觉得 'I think…' starts half of all Chinese opinions). Buddhist texts use 觉 for enlightenment: 觉悟 is awakening to truth. So 觉 spans everything from a nap to nirvana — context tells you which.
Learning-roof over seeing 见 — AWARENESS: asleep (jiào) or awake to feelings (jué).
睡觉 (sleep) and 觉得 (to feel) share one character with two pronunciations — the most common two-reading character beginners meet.