Do you mean
the American Civil War (the one fought OTL 1861-65) or just any sort of American civil war? Your wording seemed a little ambiguous to me, but assuming the second, the most straightforward way to do it (as I understand it) would be to suddenly knock out centralized and legitimate political leadership without a clear path of succession, allowing regional or ideological factions a chance to rise up and try to fill the ensuing power vacuum.
How you go about establishing this power vacuum is a little trickier. The easiest way would be to have a sudden natural disaster (the Tunguska bolide goes a bit off-course and explodes above Washington or what have you), wipe out central leadership, but a POD on that scale would probably qualify as ASB. You might also be able do it via ordinary man-made death and violence, but killing off everyone (the President, Cabinet, Congressional leadership, etc.) with no warning is kind of tricky with pre-20th century technology. Still, the earlier the POD and less sophisticated communications are, the slower news will travel and the greater the chance for miscommunication and chaos following a successful decapitation of leadership (by whatever means), and thus the odds are better that there will be multiple 'Provisional Government of the United States' claimants by the time the actual legitimate successor government (if it exists) can reassert itself. Alternatively, you could have a situation where, despite the pre-conflict government still surviving, it is viewed as illegitimate and unrecognized by all the other claimants for any given reason (i.e., it's the result of a coup or seemingly rigged election) and so the claimant factions fight against it for betraying or abandoning the principles of American governance.
Despite its. . . other issues, I think
The Falcon Cannot Hear did a good job of portraying the sort of situation OP described, where several of the major factions claim legitimacy as successors to the American government in a country-wide civil war.