-Draw your background.
-Right click on it, Convert to Symbol > Movie Clip
-Put it on the side of your "Stage" (the white area)
-Make a "Keyframe" on something like frame 80 of the "Layer" with your background MovieClip.
-Move your background to the other side of the Stage.
-Right click in between Frame 1 and Frame 80 and "Insert Motion Tween"
-Ctrl + Enter to test play it.
If it turns out right, your background Clip should move from one side of the Stage to the opposite. If the background movieclip is something simple like a large solid green rectangle for grass and a blue rectangle for sky (add a simple tree to test with), then it should "loop" once it gets to the end.
There are a lot of ways to accomplish a scrolling background effect.
You can make a large donut shaped movie clip on it's own layer and set it to "rotate clockwise" infinitely. That seems like it would work..
Something to keep in mind when making a background is that if you don't want it to end up looking like a side scrolling platform game for Nintendo, and more like a movie, then you'll need to focus on more than 1 scrolling background image. Maybe make 3 separate backgrounds and have them scroll at different paces. It's a really nice effect.
Regardless, good luck, and if you figure out a really good way to do it, make a tutorial out of it or something. :)