I successfully managed to stream to Twitch using avconv from a Debian Wheezy system.

Here is the command used to display (part of) the desktop:

avconv -f x11grab -s 1680x1050 -r 15 -i :0.0 -c:v libx264 -pre fast -pix_fmt yuv420p -s 640x480 -threads 0 -f flv "rtmp://live.twitch.tv/app/live..."

Which means: grab the top left 1680×1050 pixels of the desktop at 15 fps and encode it using libx264 with the “fast” preset and the yuv240p picture format, resize this to 640×480 pixels using 1 CPU core and format all of this as a FLV stream to be sent using RTMP to the Twitch server (using the secret live streaming key).

Adapted from the FFmpeg streaming guide.