What should I look for in a budget-level microphone if I'm going to use it primarily for capturing clean, guitar amp tones?
Can such a microphone capture both clean and distorted tones well?
Good directionality is important, so a cardioid or supercardioid pickup pattern - you don't want to be picking up spill or other instruments around the cab. Being a dynamic mic (as opposed to condenser) is a plus in some live scenarios, as you can hotswap should you have a problem with it or a cable without worrying about damaging any gear from phantom power spikes. Also, ensure it has an XLR connector - and check its sturdiness, you may encounter problems with the internal wiring if it's too weak. A Shure SM58, for example, when unscrewed has really solidly constructed internals; everything is literally either hot glued together or screwed and fitted as tightly as it can be to avoid loose connections developing from being thrown around both by artists and roadies.
What's your budget? You can get good value, you can get cheap and you can get DIRT cheap. The quality curve is inversely proportional. ;-)
Unless you've already heard a mic it may have an awful response with regards to transients, or its characteristics may be very uneven - the frequency response as stated on boxes of cheap equipment never quite matches up, so an evaluation (or no-penalty return agreement) is essential for mics. Ideally for easy stage placement, a side-facing diaphragm is preferable, as you can mount it in a mini-stand or (not always advisable!) hang it over the front of the guitar cab with an XLR lead. Not recommended, as it puts great stress on the cable and mic connectors, but sometimes when you need "that sound"...
Unfortunately you'll never get the best of both characteristics - build quality and sound quality - I've personally used SM58s, SM58 betas and both Sennheiser E609 and E906s on guitar and bass cabs, and my favourite so far has to be the E906. The tone I got from a bass cab turned up to 11 in a small venue with the 906 was outstanding, the plus being that both Sennies are specifically designed for easy guitar & bass cab placement. I also felt that I got a better, cleaner and more defined raw sound from the Sennheisers than even the more expensive Shure. Spill and directionality were vastly superior, and getting a raw good sound from a well-designed mic is three quarters of the battle won before you even start.
There's a very helpful YouTube video comparing both E609 & E906 side by side at http://www.youtube.com/watch?v=8yrrVvoXJSQ&feature=related if you'd like to internet audition.