I don’t get along with that guy.
sich mit jemandem verstehen = to get along with someone
Though I think I’ve more often heard “mit jemandem gut klarkommen” for this purpose in German:
Ich komme mit ihm nicht gut klar.
Or maybe: Ich komme mit ihm schlecht klar.
“klarkommen” is more like “to know how to handle” or “to cope with”, and is used more often with situations or tasks and less with people, and if it’s about people, it’s people that you are forced to get along with, such as co-workers, that you didn’t pick a friendship with. When two people “miteinander klarkommen” it means that there’s no fighting, but not necessarily a friendship.
“sich verstehen” is more friendly. It’s used when people think alike. It’s more than just getting along and each doing their own business. The negative “sich nicht verstehen” on the other hand is indeed similar to “not getting along”, but it doesn’t mean that there’s constant fighting.
Mit diesem Typen verstehe ich mich nicht. - We don’t think alike, but as long as we mind our own businesses there’s no problem.
Mit diesem Typen komme ich nicht klar. - I don’t know how to cope with this guy. He gets on my nerves every day, and I’m frustrated.
Of course, the lines are blurry again, and you may see one expression used in place of the other.