No, you don't need to watch the first part to fully understand the story, characters and themes of the second one.
All you need to know is that the main character is a hitman specialized on making his kills look like accidents and that he's out of business and believed to be dead. But as much is already conveyed right at the beginning of Mechanic: Resurrection. There are no other story elements or characters from The Mechanic who make an appearance in the sequel. There is some additional backstory and characters from Bishop's past life involved in the new film, but none of them are mentioned in the previous film.
All you need to know about this film is that Jason Statham plays a classic Jason Statham character and goes full Jason Statham on people. ;-)