I'm actually moving up to JC Heights in two weeks. Not sure of where the closest trails are, but there's got to be lots out just a bit northwest of Hudson County. I also run a lot (trails) and will be looking to do a lot of that as well. Not sure if Liberty State Park or any parks in Hudson County have any dirt/gravel running paths.
Born and raised in jersey city. I use to live in the Heights as well. My mom still lives there and dad is in a nursing home near the square.
No matter where you drive to, you are like an hour from some good riding spots like Lewis Morris, Chimney Rock, 6 Mile Run, Hartshorne and even Ringwood. Since the summer is approaching places like Hartshorne and even 6 mile run, it may take longer than an hour with summer traffic, so keep that in mind.