Are you able to become Jarl in any of the given cities?

I've had a fascination with the different titles you could aquire and was wondering if one could possibly become Jarl in, say Markarth, Solitude, or Whiterun?

It is not possible to supplant any of the existing NPCs as Jarl in the base game. (I wouldn't put it past someone having created a mod to do so, however)

The closest you can get is to become "Thane", which you can do for each of the 9 holds, and will earn you a Housecarl (a la Lydia) and let you clear your bounty from the relevant hold guards, once.

Vanilla-wise, there is no way to become Jarl of any city (as unfortunate as that is).

If you use mods there is a specific mod for Skyrim (not SE), that allows you to become the Jarl of Ivarstead (which isn't a major city, but allows you to simulate a Jarl-like lifestyle)

There's no way in vanilla. However, there is a mod to become High King of Skyrim. This would then force all other Jarls to report to you...the Jarl of Jarls.

(Oldrim) Become High King of Skyrim: http://www.nexusmods.com/skyrim/mods/41087/? (SSE) Become High King of Skyrim: http://www.nexusmods.com/skyrimspecialedition/mods/2152/?