Sorry, to the best of my knowledge you cannot play online via Xbox live without Xbox live gold. I remember way back in the original Xbox days, people used to use virtual LANs to play online without Xbox live gold, much the same how PC users do with Game Ranger or Hamachi. Nowadays though, games don't generally support LAN play and those that do have a ping limit specifically to prevent this kind of thing. It might be worth a shot if you have a few friends within a few hundred miles. Just don't expect server browsers and matchmaking.
But you don't necessarily have to pay for a gold account. Microsoft offers a bunch of different ways to "earn" free gold account time. For instance, if you use Bing, each search you do accrues points, and 7000 points can buy one month of Xbox live gold. There's also a bunch of ways to get free trial accounts, but they're only good for a month and last time I tried it you had to make a new account for every trial.