But don't fret; you're trying to obliviate the old box, not rescue it, so you can't really make a mistake!
The data you're concerned about resides on the computer's internal hard drive. Smash it.
How to get to it? Make sure the computer itself is not plugged into the wall, and then turn every screw on the case (counterclockwise), push anything that looks like a latch release, until the case opens up. (Some screws may be hiding inside the case's rubber feet. Find 'em, turn 'em.) Don't be afraid to force things to identify what might still be holding the case together. Wear gloves if you're ultra-cautious.
How to identify the hard drive? There aren't many metal-encased assemblies inside; probably just a power supply, the hard drive and maybe some disk drives. (Disk drives will have some access to the outside world; a slot where you can insert and remove disks, CDs, DVDs, or whatever. Anything with such external access can be ignored, as can anything that's flat and green and festooned with little electronic gizmos.) If in doubt as to what's what, keep turning screws until all the assemblies become unmounted. Go after the screws that hold the metal-cased assemblies together, unplug whatever cables you can (and cut what ever you can't). When you've got the thing stripped down as much as you can, whack whatever looks whackable. Use a hammer.
If something on one of the assemblies identifies it as the hard drive, you can concentrate your efforts on it. (There's no security advantage to disembowling a power supply, but if you don't know which is which, there's no harm to it, either.)
Some computers have more than one internal hard drive, typically two. Get 'em both.
Your locality may have some ordinance about disposing electronic waste, of which you'll have quite a pile when your done. Be sure to comply.
Have fun!