In Slackware, hibernation works out-of-the-box and can be easily done from KDE's menu. Resuming from hibernation, however, is not set up, so when you turn on your computer again, it starts as normal. That is, if your hardware supports it and if you've set up a swap space larger than your RAM.
To tell your kernel where to resume from, you need to write the info to lilo.conf:
Replace /dev/sda6 with your actual swap space. Swap partitions are perfect, I think swap files should work too. If you don't know where your swap lies, look at /proc/swaps.
The above is everything I needed to do with my Slackware64 13.1 on a new laptop. I hope that works for you as well, but there are many requirements and intricacies not mentioned above. Google might help you. Please edit this document accordingly if you find out more.