Is this bike new to you? Did it ever run correctly so you know what it should run like, or is this possibly your first inline four?
The reason I ask is that many first time inline-four owners are not exactly sure what to expect, so when I work on their bike, it will be way out of whack, as you said. By that I mean it will only be running on two cylinders and those two will be way out of sync, and sometimes it will actually be made to idle on the main circuit (instead of the idle circuit), but the owner just thought it needed a minor tweak or two.
To expand on what Texas Kz said...
So if this bike is new to you, you need to do a compression test with a good tester. Not a harbor freight tester. They often read way incorrectly. The compression should read near 180 psi. I've never seen a 550 read much lower than that regardless of miles or condition.
If this is your first bike ever, a valve adjustment will be tricky. Normally I would say it needs to be checked and adjusted as TexasKz said, but for now, we want to avoid putting in any more bugs, and it will usually run ok as long as it doesn't get too hot if the valves are just out of spec. If you do open the valve cover, read the valve train warning in my signature to prevent doing damage.
s3.amazonaws.com/gpzweb/CamsKZZXZR550500...KZZXZR550500400.html
Next is to confirm ignition. I suspect you are probably having ignition and carb issues. Carb issues are the most complicated, so it's better to take care of the easy stuff first.
A first test would be a water spray test on the exhaust. Get a squirt bottle with water, an actual toy squirt gun works best, probably. When the bike is cold, start it up and use the bottle to test each header to see which ones get hot right away, and which ones only get warm. The water should dance on the metal without really getting the metal "wet". That will tell you which cylinders are getting combustion. You know you are getting spark on at least those cylinders.
To check for actual spark, take out all four plugs (this should be done on the compression test as well, so it's a good time to do both), connect them to the wires, and make sure all four plug bodies rest in contact with the engine. Crank the engine over and confirm you have spark on all four. It's much easier to do this in a garage or at night.
Make sure the advancer mechanism is free to move and snaps back easily. That will cause strange running if it doesn't move freely. To confirm the advance timing, you will need a timing light.
Most of these test are described in the manual. Here's a link to several. You should probably grab all of them while they are available.
s3.amazonaws.com/gpzweb/Mnls/Mnls.html
Here's a link to a site about your carbs. It's not meant as a substitute for the manual, it's for details not covered in the manual.
s3.amazonaws.com/gpzweb/TK22mainPage/TK22mainPage.html