Old English felan "to touch, perceive," from Proto-Germanic *foljan (cf. Old Saxon gifolian, Old Frisian fela, Dutch voelen, Old High German vuolen, German fühlen "to feel," Old Norse falma "to grope"), from PIE root *pal- "to touch, feel, shake, strike softly" (cf. Greek psallein "to pluck (the harp)," Latin palpare "to touch softly, stroke," palpitare "to move quickly"), perhaps ultimately imitative.
The sense in Old English was "to perceive through senses which are not referred to any special organ." Sense of "be conscious of a sensation or emotion" developed by late 13c.; that of "to have sympathy or compassion" is from c.1600. To feel like "want to" attested from 1829.