No, Lebron, while an excellent athlete and someone to be admired, is not the best, because Michael Jordan was still the greatest athlete of all time. No one can eclipse Michael Jordan for what he accomplished in his sport when he did. Tiger Woods is also a better athlete than Lebron, although Lebron is a more ethical person.
Lebron James makes headlines for his life both on and off the court, but that does not make him the best player in the NBA. Indeed, players who are famous usually become that way by being outspoken and drawing attention to themselves, not for being particularly skilled at their chosen sport. Compared side-by-side, there are plenty of other players with more respectable public personae who play the game just as well as Lebron.