You can't directly check your CDI without a known good one to check it against. It's a process of elimination, where if everything else fails, your CDI is probably shot.
The 550 ignition system, being CDI and not TCI, is very simple. You don't need the reg/erct, battery, lighting circuit, etc. All you have is the generator, cdi, coil, plug, ignition switch and kill switch, the rest is entirely separate. Forget the battery circuit for the moment, it's utterly irrelevant.
You'll need a multimeter, and an electrical schema wouldn't go too far amiss either - here's one in italian - http://digilander.libero.it/XT_550/I...co%20XT550.jpg
First thing to do is to check the kill switch. All sorts of crap can get in there, they can block up and fail. It works by earthing the CDI, so the easiest way to test it is simply to disconnect it entirely. Check for continuity between the kill switch contact (where the black and white wire runs in) on the socket and the frame. continuity when in "off" position, none when in "run".
Now the ignition switch (key). Check for continuity on the black and white wire running in and running to earth. Again, only continuity when it's off / lock / park
Now check the wire feeding the kill switch / ignition switch from the CDI. There should be continuity between both ends (obviously), and *no* continuity from either end to earth. This is the black-and-white wire from the CDI.
All good? Time to check the "hot" side of the coil. remove the sparkplug, and reattach it to the plug cap. check for continuity between the centre electrode and earth (should be) and the outer electrode and earth (shouldn't be, as long as the plug is away from the motor. Common failure here is HT lead disconnect from plug cap, shot contacts in the cap itself, shot cap, bad connect from coil to HT lead, bad connect between coil and earth.
Test resistance of hot side of coil (resistance between orange lead into coil and plug electrode) against spec.
Test resistance of cool side of coil (orange to black) against spec.
all still good? Test continuity between orange from CDI to orange in to coil. should be good, with no continuity to earth.
Now check the generator. continuity between red and brown from generator, and resistances good. No continuity between red or brown and earth.
you can't really check the advance detector without a strobe, so plug and unplug repeatedly. contact cleaner if you have any. it's the 3-way connector from the generator with white/red, white/green and green wires.
If that all still checks out, it could well be your CDI that's shot. double-check the earths (black wires) on the CDI to be sure, and all earth connections to the frame, but that's about all you can check.
You can use a non-yam CDI to check, the easiest one to use is a C50/70/90 CDI (and has the benefit of being cheap as chips). This has no advance, so you connect to either the white/red or white/green pulser and the green, plus the red and brown. If you connect the wrong pulser your ignition will be too advanced, and it will throw you off when you kick it over. if you connect it to the right one, the bike should run, but you will only get about 3.5-4K RPM tops. This is all a last resort, though.
FWIW, my money is on something being earthed that shouldn't, or a bad connection on your HT lead.