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 carefully.