How is Paul back? did they ever catch that he killed that one neighbor in season 1?

paul went to jail for the murder of martha hubers sister: felicia tilman. felicia faked her own murder and since paul had attacked her a few hours prior, he was the main suspect and was arrested. he did time in jail, which we can see when mike goes to jail for the murder of monique.

hes back because he was arrested for the murder of felicia tillman ( martha hubers sister ) because felicia set him up but one day she was cort speeding and then they let paul young go cus they found out she set him up and shes alive .

The whole point of Felicia setting Paul up for her 'murder' was because he got away with killing her sister Martha(the neighbour from series 1).He's out because she was caught speeding and they discovered her real identity. So technically he's innocent of the crime he was convicted for.