The safest way is to get an external enclosure.
If you're going to use the old power supply, make sure the power supply and the laptop are plugged into the same wall outlet/power strip -- this ensures they have a common ground.
Do not leave the laptop running on battery power.
As Chris mentioned, the old AT style power supplies have a manual power switch, but no SATA power connectors. You'll need a Molex-to-SATA-power adapter to power the drive.
If your power supply is an ATX/ATX-2, it won't power on by itself. Normally, the power-on signal comes through the motherboard's main power connector when you hit the power button on an ATX system. You can force the power on by connecting
pin 14 (PS_ON) to one of the
GND pins (13 or 15, for example). (Diagram source and more info)
Be very careful when using an ATX power supply in this fashion. This is meant for testing a PSU, not extended operation.