Without a separate background light, it may be hard to control the look of your background. One thing to be aware of, is the inverse square law, as it pertains to diffuse reflection (which is what you'll have from a backdrop). The ISL states that something that is farther away from the source of the light, will reflect an amount of light that is inversely square to the distance. For example, if your subject is 2 feet from the light, and the background is 4 feet from the light, the background will reflect 1/4 the amount of light that the subject would. If the background is 10 feet away, it would reflect 25 times less light than the subject.
So you can't just hang a backdrop of a certain color, and expect it to photograph just how it looks to you...you need to consider distances as well.
ALso, why should I avoid mixing ambient light with window light? I have a nice huge window in my living room, but it's not enough light so I wanted to put a light on the opposite side for fill.
Color temperature. Light coming in from a window will have a certain color cast to it (it's color temp or white balance). That's OK, because you can set/change the white balance setting on the camera to match the light, and thus get accurately colored photos. But if you turn on an interior light/lamp etc. it will very likely have a different color temperature than the window light. So now, if you're WB is set for the window light, the house light will cause a color cast. If you set for the interior light, the window light will cause a cast. And if you set it somewhere in the middle, they may both cause an inaccurate color cast.
It's just not a good idea to used mixed lighting, unless you are going to convert to B&W all the time.