Unfortunately I live in an apartment complex with neighbors close by, preventing me form being able to play my amp without plugging headphones in. The sound quality is mediocre at best, and I am unable to play with distortion on or else the sound is way too loud through the headphones. Any suggestions on how to get a much more playable setup with headphones? I have a pretty pathetic practice amp from when I tried playing years ago that I use now, but I am willing to upgrade to a better amplifier if it will make that much of a difference through the headphones. Probably 500$ or less i would hope (No need to go all the way to 500$ though!).
1. You need a good set of headphones. A four-buck set of earbuds ain't that. Decent studio-quality over-the-ears are going to be $50-100.

2. Are you sure you're plugging into a HEADPHONE jack on your current amp?

3. Why not pick up a used Pod XT (probably $80-90) bean to use as your headphone source?
my tech mates at the music suggested that cheap lo-fi headphones come closer to speaker type tones than hi-fi ones. So I'm using a set cheap Ashton electric drum kit headphones.

I'm wondering about the volume problem. If the amp has a headphone out, it will be solid state, so can't you just turn everything down and still get a reasonable tone? Otherwise you might have to go clean plus pedals. - Or buy a better amp.
I have a Korg px4. Tascam made wireless infrared headphones specifically for this unit, but they were way more than the unit itself, when I first got it, so I kind of ruled them out. I just found a brand new pair about a year ago on ebay for $35. They are awesome together. Anyway, this or a pod is going to give you a better tone with headphones than an amp, plus they have lots of other features that are cool.
First get the best pair of Audio-Technica Headphones you can - the ATH-M50x are very nice.

Second if you want to reuse the gear with your amp then get a POD. With your budget a used HD500X is in scope.

If you don't want to reuse the gear with your amp look at amp simulations (VST) for your computer. I suggest either a Rocksmith video game cable or an Unbranded Guitar to USB interface from Amazon ($20ish) and Peavey ReValver software (free to start and only license what you want - ValveKing is free).

So realistically you can do this for about $20 and upgrade the headphones when/if you want to.
Jackson Kelly KE3 - MIJ (Distortion/Jazz)
Jackson DKMGT Dinky (EMG 81/85)
ESP E-II Eclipse Custom (JB/'59)
ESP LTD EC-1001FR (EMG 81/60)
Fender MIM Strat

Mesa/Boogie Dual Rectifier Roadster 212
Laney IronHeart IRT-Studio
Peavey Vypyr 30
Peavey ReValver Amp Sims
TOOOO many T.C. Electronic Pedals. . .