If you're using high purity isopropyl alcohol, the advantages are that it is non conductive (being a non polar solvent), is not terribly reactive with the things you'd normally find in a computer, and generally dries fast. Water is a polar solvent, so you don't want to be using it in computers in larger quantities. 70% seems to be what people use, but you really want to be using 90% or better.
It's a very effective cleaner that at the quantities you use won't end up poisoning you, or dissolving your fingertips.
It's apparently safe to immerse entire devices in isopropyl alcohol—it seems to be a common way to save 'dead' cellphones—but I wouldn't recommend it as you'll see. It probably won't damage electronics; this thread on the HP calculator forums talks about immersing an entire calculator in alcohol, though with no follow up on whether it works.
via here - no idea on the original source.
And this is allegedly an entire PC soaked in rubbing alcohol. (No, do not do this. isopropy alcohol is highly inflammable - it will evaporate into something that can be ignited at low temperature (13-15 degrees), and will catch fire at around 400 degrees C, and we do not want to be responsible for house fires. BAD BAD IDEA.) And since this is kinda being mentioned in the comments (Its potentially explosive under the right conditions - I do not recommend trying to blow up plastic water jugs unless you're a trained professional and/or a chemistry teacher).
It will, however, probably dissolve things like thermal paste (which can be conductive, especially in formulas that contain metal.) As such, you don't want to be using it in bulk (by spraying or dipping a whole panel.) Additionally, it's a bad idea to combine it with grease, since you might end up dissolving some minor thing—conductive stuff in CPU sockets is bad, as would be accidentally removing all the lube from the mechanical parts of a drive. It'll also loosen any stickers present on the hardware. Apparently it also does nasty things to plexiglass and Lucite, so test before you clean any plastic bits.
Applying it to a paper towel may leave lint (coffee filters seem to be a common lint-free alternative), and the computer has lots of bits that stick out which may catch and rip out bits of it. I've generally used the stuff for spot cleaning on a cotton swab for particularly nasty, hard to remove dirt. It's also pretty good for cleaning contact pins or dirty lenses on optical drives (often using a lense cleaner) VERY carefully. As such, while alcohol itself might be safe, be very sure to clean the surfaces again after you have removed the intended residue with something lint free.
You'll also want to be careful with a cleaning cloth, tissue or other cleaning material not to bend anything. Bent pins are a PITA.
There are also large, pre-moistened wipes—there's variations for screens and other surfaces—that also contain other substances. These are excellent for external cleaning, and are lint free.
In most cases, air dusters for electronics and a dry microfiber cloth for surfaces work for me. Save the alcohol for tough situations like cleaning off heat sink compound or stuck-on dirt.
TL;DR: "How do I clean a computer with isopropyl alcohol?"
Sparingly. Use it when you need to clean electrical contacts and need to remove tough-to-remove substances. It's tricky for bulk cleaning but works well with specific issues. It's definitely the thing to use when cleaning monitors, degreasing a touch screen (ewww) or for cleaning dirty contacts. In many cases an air duster may be safer since you aren't touching the contacts.
Sources:
wikipedia - for the chemistry bits.
