Is /var on your root filesystem? You could try using xdu to see where the disk space is. If
/var is on the root fs, try doing an apt-get clean before and mount /var/cache/apt/archives
on another disk like a USB stick for instance.
Use dpkg -l | grep ^ii | grep linux to see which kernels are installed.
Then use uname -a to see which one is in use on the system currently.
Carefully check versions and use apt-get remove linux-headers-VERSION then do
the same with any linux-image-VERSION files also not in use.
You'll get loads of free space removing old kernel versions and header files, just do it