It's a nice neighborhood and there are handful of bars or whatever nearby. But by and large Jersey City is nothing near what Hoboken is. And the problem with Hamilton Park is it's a bit of walk to the PATH at either Grove Street or Newport so it's not especially easy to get to Manhattan either. If you're okay walking 15 minutes to downtown JC you'll find a few places that are good. Down by the waterfront and in Paulus Hook there's a few bars there like Taqueria and Zeppelin Hall down at Liberty Harbor that are good spots but that's a good walk, possibly an Uber from Hamilton Park. There's also some rooftops along the Hudson but again that's a hike. If you're dead set on Hamilton Park I guess you could walk through the mall and hit the light rail to Hoboken (sort of a weird walk back at night though) and that's still maybe 15 minutes.

For reference I lived in Paulus Hook down by the marina which is a wonderful neighborhood, but the light rail goes right through the middle of the neighborhood so it was easy to get places and Grove Street PATH was only a five minute walk. Even though the Paulus Hook bars aren't exactly raging and close often before midnight the ease of being able to get either to Manhattan or Hoboken made it worth it. I could be in the heart of the village in 15 minutes because I was so close to the PATH. And price wise it's not really any more expensive than Hamilton Park area at the low end at least, and a lot quieter.