A proper name. Several bearers of it are known, including several writers -- notably an (?)Alexandrian historian, reckoned to be the source of the so-called vulgate tradition on Alexander the Great. See generally FGrH 137; OCD(4) s.v. Cleitarchus (p.330); and under epsilon 3995.
When Cleitarchus lived and wrote is a matter of dispute. Orthodoxy assigns him a "high" (a.k.a. early) date, in the last years of the fourth century BCE or early in the third. A fragmentary papyrus published in 2007 appears to state that he was the teacher of king Ptolemy IV Philopator (pi 3039), but Prandi [below] does not consider this enough to overturn the accumulated evidence and probabilities for an earlier date.