This was discussed on page 1 of this thread, albeit, not with a lot of detail.
There are three principal options:
1- HDTV Antenna: A decent antenna, with an amplifier, with provide you with an excellent 1080 signal and all the bells and whistles for surround sound. Relatively cheap to buy, it may or may not be practical for you to install (or even possible if you are in an apartment without a balcony/proper window facing).
Given that it also supplies you with live programming for basic channels and access to basic news and live events (Oscars, SuperBowl, Olympics, Major Sports Playoffs, etc.) with excellent picture quality and with solid and consistent reliability, a good HDTV antenna offers great value. Set-up cost is a one-time affair. After that, FREE is as good as it gets.
I have a very good HDTV antenna (Clearstream 4) and I can recommend it without reservation. You probably won't need as high-end a model as that in Brooklyn - but if you want a recommendation I certainly can confirm it works reliably and VERY well. It gives me 22 channels in Toronto. I linked to the antenna itself, but you will also need a mount, an amplifier, cable, brackets etc. to complete your installation.
There are, doubtless, installers in your area who will, for a fee, show up at your home with all the gear and tools required, crawl around on your roof and just magically install all of this for you. That approach to just biting the bullet and getting it done has a lot going for it as a one-time fee, imo. As I'm not much of a handyman, that's what I did. YMMV.
2: XBMC Mashup Streaming Live Sports: (Pirated Signals) Casual sports fans can use XBMC and live streaming services through Mashup and Sports Devil etc. to access live streaming of sports programming, including the NFL. Setting aside the legalities of this approach, I personally have some difficulties with relying on this for far more practical reasons: reliability and quality of picture. Because the pirated streaming signals for these games degrade significantly in proportion to how many people are accessing the signal at the same time, the upshot is that the more important the game is to you, the less reliable your access to it becomes -- and the worse the picture quality is.
As a sports fan, this is unacceptable to me for the sports that I care about. As I don't care much about basketball, I can go this route for the 5- 10 basketball games I might watch in a given year. Maybe you feel the same way about basketball as I do, or you feel that way about hockey, say. If so, this is an acceptable approach for those sports you are only an infrequent and casual fan of.
As you are asking about the NFL, specifically, and you mention recording a game at the same time as you are watching another, you clearly aren't a casual fan of the NFL. This issue matters to you and you care about watching your games frequently. As you are probably both a Jets AND a Giants fan, I guarantee that the problems of relying on streaming pirated signals will prove to be a real issue for you when trying to watch those teams in Brooklyn. A pirated signal approach is just not going to work for you, imo. Your main options are HDTV antenna or NFL Game Pass (preferably, both).
That doesn't mean that XBMC isn't awesome. It is extremely awesome. But where it shines brightest is in access to streamed pre-recorded programming in a manner identical to Netflix. Live streaming is available over XBMC too, it simply isn't its forte. You should be able to use it for access to sports highlight shows on ESPN, etc. with very decent reliability so there is value in XBMC for some frequent sports programming.
NFL Game Pass: NFL Game Pass $220/yr. NFL Game Pass is a live streaming service officially licensed by the NFL. It has a number of bells and whistles that come with it, but on the whole, what it offers is access to Pre-Season and Regular season games. Playoffs are not offered live in the subscription version. Given that all NFL playoff games are reliably available for free over a HDTV antenna in the USA and where most Canadians live, a combination approach is required here. So yes, you really will need a HDTV antenna.
The main problem with NFL Game Pass is not the quality of the product (it's pretty good), rather, it's the cost of the product, especially when compared to MLB.TV and the grey-marketed services Hockeystreams and Ballstreams.
It looks even worse when the reduced number of games are taken into account. MLB.TV is running ~15 live games a day, every day, for six months+. Hockeystreams and Ballstreams offer about half that number each - but it's still a helluva lot of hockey and basketball (with Hockeystreams, the other league games they offer more than makes up for it).
In contrast, with NFL Game Pass, it's 16 regular season games for each team and a handful of pre-season games. And that will cost you $220 a year for the basic subscription. So the service that offers by far the fewest games to watch is also, by far, the most expensive to buy. It's a poor value in comparison to similar services for the other major sports - and that's why I complain about it.
That said, it's probably the only real option there is for you if you want a high quality HD picture/sound and reliable access to your games.