Old English climban "raise oneself using hands and feet; rise gradually, ascend; make an ascent of" (past tense clamb, past participle clumben, clumbe), from West Germanic *klimban "go up by clinging" (source also of Dutch klimmen, Old High German klimban, German klimmen "to climb").
A strong verb in Old English, weak by 16c. Other Germanic languages long ago dropped the -b. Meaning "to mount as if by climbing" is from mid-14c. Figurative sense of "rise slowly by effort or as if by climbing" is from mid-13c. Related: Climbed; climbing.
1580s, "act of climbing," from climb (v.). Meaning "an ascent by climbing" is from 1915, originally in aviation.