Brooklyn/NYC people: How do you compost?

OK, so this is my second year of SFG'ing. I have a 48 sqft. 4x12 bed which takes up a good chunk of my backyard. Last year I bought a bunch of different composts to make my mix. My garden thrived, though I started late so most of my crop did not give me much before it got too cold. I feel my mix was good, though, as the plants seemed healthy.
I had enough leftover compost from last year to get me started this year but now don't have any left. I want to start my own composting but am a bit overwhelmed with all the choices out there and the limitations of my urban location.

Anyone in Brooklyn or nearby with limited space have any success stories or advice?

For people not familiar with the area, here are my concerns:

I can afford to take up maybe about 6-8 sqft. with a compost setup. Definitely no more than that. I can probably make space on my (crappy, weedy) grass or on a paved area just behind my house. Either way it would probably be exposed to about 4-5 hours of sun a day. I also have a dark, dingy part of my basement I could set something up in if that would be better, though this area is probably nut much more climate controlled than the outdoors.
Aside from space, another main concern is the temperature range here. We go from many sub-freezing days in winter to many humid 90+ degree days in the summer.
The native soil also has a rather high lead content (over 1500 ppm) according to a test last year. For this reason I assume I should not compost leaves and grass from my yard? Is that correct?

Due to these issues I am leaning toward a small to medium pre-made plastic type of composter as I don't have room for big piles. I figure I don't need anything that big as my garden is only 48 sqft and I won't be putting tons of leaves and clippings in there (or do I need to?).
Should i go for a tumbler? A worm setup? Or is just a stationary barrel ok? Would kitchen/garden scraps be enough to keep it going and produce enough compost for me (It's just me and my wife, and a baby on the way)?

Every time I think I have made a decision, I read something that makes me doubt it. Looking forward to your ideas...

If you can get to Sheepshead Bay, try getting some of the wasted fish remains.
from the party boats.
Even fish some don't want.
I don't know what it's like there now, however years ago lots of good composting material was available. I was not aware of this then.
I used remains of 15 lb steel head in my garden now. Oct was the season, buried lots of stuff about 12 inches down and was all decomposed in the spring when planting.
I suppose rats & cat might be a problem also.

Delay, I also have a tiny outdoor space since moving to a condo in the fall. I brought my vermicomposting with me for indoor composting in the winter. It's a Rubbermaid tub like you'll see in other posts here on SFG forum and handles much (but not all) of my compostable scraps. For summer, I'll expand to using worm tubes right in my SFG raised beds. Here's a link, although Josh's great photos have been removed from the thread https://squarefoot.forumotion.com/t1603-worm-tube-for-the-sfg?highlight=worm+tube.

Let us know what you try in your yard!

Thanks for the replies.

Jimmy: Sounds like an interesting idea. I do have several neighborhood cats that hang out in my backyard (and even more squirrels) so I don't think that would be the way to go for me. Thanks for the suggestion, though.

Janas: Yeah, I'm leaning toward the worm composting as well. When you do worm composting, is that all you need? Can I keep it outside in the winter or will that kill the worms? Maybe I'll try the worm tubes as well. I had lots of big worms in the garden last year, so hopefully they'll be back this year.

Any other ideas, people?

Second thoughts make me believe you might just do well with a tumbler.
Being secured no pests will get in it, however that won't deter them from trying or even being in the area...( I remember well the city's problems, as we were superintendent's of an apt house in mid town Manhattan)
I have an Envirocycle tumbler which I am very happy with even tho it hasn't completed anything yet since Oct.
An added benefit of this tumbler is that the base holds fluid that's called
"rough compost tea" I have emptied it and it looks good.
The people who make this tumbler are extremely helpful and just could help you along with using it, should you decide on one.
I have 2 large compost piles, so my tumbler is only another toy for me to play with.
http://envirocycle.com/us/home