I am having difficult first year in my part-time PhD program; How can I figure out whether I should leave it or not?
I'll keep this answer fairly general, because what you describe sounds like a fairly general situation. 

The transition from undergraduate to Ph.D work is difficult because you're going from a very structured environment (take classes; get grades) to a very unstructured one (take classes, but focus on learning rather than grades; deal with material that's more cutting-edge and therefore open-ended; start getting into research, which is really open-ended). 

This is hard, and it's normal to expect a period of transition where things seem out of whack. 

Coupled with that is the shame: people going to grad school are usually the ones who do well in undergraduate and have done well all their life in school. You've received praise from your teachers throughout, and have received constant encouragement. 

Now you're in grad school. Everyone there is like you, so the competition is more fierce. There are no pseudo-objective measures like grades so there's no way to compare yourself to others (which for high achievers often means that you compare yourself negatively). And above all, all those nice professors who used to encourage you have turned into scowly curmudgeons who expect more and more and more...

It's a wonder everyone isn't depressed. And on top of that you have a full-time job !!

The recommendation that I will give, and that everyone here is giving you, is to ease up on yourself and try separate out your feelings of shame and inadequacy from the reality. You should find a support network of friends, hang out in study groups, do extracurricular stuff. The friends you make in grad school will be friends for life - such is the nature of this crucible. Since you're working full time this is going to be even harder, but maybe you can find others who also work full time and are in a similar position. Also see if you can find senior grad students who are part-time: their experience will be very valuable. 

And then in a year you'll be back year asking about how to avoid discouragement in research, and how to avoid procrastinating, without realizing that this is a sign you've made it through the first year successfully :)