I've done a version of this in a couple of games with the thought that in a world with prevalent magic (as opposed to a couple of relics that he happens to have encountered in his life), Indy would at least have a smattering of it available.
My build has been Str based Variant Human.
- Indy really needs good Str/Dex/Con/Int/Cha based on the fact that he wears no armor, never gets hit, mostly punches, but is still a good shot, takes a TON of licks and keeps on kickin', knows his stuff about history, etc, is charming as hell, but wisdom... not so much. He's pretty rash and often makes poor decisions that end up working out.
Human Skill: Investigation
Human Feat: Tavern Brawler (I mean, come on... Indy punches a lot, the whip and gun are just there).
Background: Archaeologist
Skills: History, Survival
Feature: Historical Knowledge
Class(es): Rogue 1/Fighter (Eldritch Knight) X
Expertise: History, Investigation
Skills: Athletics, Perception, Persuasion, Sleight of Hand
Feats for later: Lucky (because really...), Ritual Caster, +ASI