FCP-How to export the background transparency?

Do you just want to export a still? .gif is usually a still image format, although it can be used for simple animations, not out of FCP though as far as I know (never tried).

If you want to export the entire clip you need to export it in a format that supports transparency (alpha channel). Most footage edited in FCP is DV footage, am I correct in assuming that you are editing footage shot on DV. The DV codec does not support transparency.

Find a codec that does, ie the Animation codec (there are probably others). When you export it, there will be an option for colors, you need to select millions of collors+, the plus means millions of colors+an alpha channel.