Welcome aboard!! Wow! The C1 is the first year of the "Custom"; model year 1977 although I'll bet yours was made in late 1976. I have the same bike. That's a really low vin number. That bike was probably made the first month of production. Take a look at the steering neck to see if it has a white label. If it does, what is the date of manufacture? Here's some advice for what it's worth, and below that are a couple pics of what a stock bike looks like that I hope will help.
Many newbies and not so newbies run into the same problems or issues with their bikes which could be easily avoided with a little preparation. Here are a couple recommendations that I hope you will find useful.
Before riding your KZ you should take a little time to become familiar with it. Simple things like checking the oil, tire pressures, etc. are explained in the Kawasaki Owner's Manual. If you bike didn't come with one (they normally are in a little holder on the bottom of the seat) you can usually find one on eBay - be sure to get one for your exact year/model.
If you plan to maintain your bike yourself rather than rely on a shop for all maintenance, you really should buy the Kawasaki Factory Service Manual for your bike. It will save you time, money, and frustration by enabling you do perform tasks correctly the first time without damaging anything. No one has ever regretted buying one of these. The factory service manual provides instructions on the various maintenance procedures and schedules for your bike, procedures for diagnosing and correcting problems, proper torque values for the various fasteners (VERY IMPORTANT), and insight into how each system on your bike works even if you are not repairing that system. Some tasks that can adversely affect the safety of the bike, such as installing the front axle, are not intuitively obvious and are nearly always done wrong unless the manual or someone who has read the manual is consulted. Studying the manual before you attempt performing service on your bike will impart an understanding of the system you are working on which will increase the odds of success and safety.
The manuals usually are readily available for most models on eBay and other sources, but be sure to get one that has your specific year and model in it as there are differences between the various KZ650 models. I do not recommend Clymers, Haynes, or other aftermarket manuals as I have seen instances where they provide inaccurate advice. You may decide to use those to supplement the factory manual, but in my opinion they are not a suitable substitute. If you have trouble finding one for your model ask us for help.
If you plan to do your own maintenance you will need some tools that you may not currently own. Kawasaki uses numerous steel screws and bolts many of which are threaded into the aluminum engine. The bolts are much tougher than the aluminum and if over tightened will strip the aluminum threads. Although these can be repaired, preventing damage by applying proper torque will save you grief, time, and money. Tightening fasteners properly requires torque wrenches. Various vendors, including Snap-on, CDI, Sears, etc., market torque wrenches. The two most popular types are the bar type and micrometer type. Either will work, but the micrometer type is a bit easier to use, especially if you are tightening several fasteners with the same torque - such as cylinder head covers. I have found that it takes 2 different scale torque wrenches to properly torque most if not all the fasteners on the KZ650. There are many fasteners that require low torque. These include such things as 6mm oil pan bolts, cylinder head cover bolts, etc. for which the torque in 61-78 inch pounds. These low torque bolts require a wrench that is accurate at low settings and this wrench should be graduated in inch pounds. In addition, there are numerous high torque fasteners such as the front axle nuts (51 - 65 foot pounds) or swing arm pivot shaft nuts (58 - 87 foot pounds). These require a wrench that is accurate at higher settings; this wrench should be graduated in foot pounds. Using the proper torque wrenches to correctly tighten fasteners will help prevent damage and contribute to a safe bike.
The crosshead screws on the Kawasaki that look like "Phillips" screws are not Phillips; they are Japanese Industrial Standard (JIS). Using a Phillips screwdriver tip on these will damage the screw heads. JIS screwdriver bits are available, but if you don't want to spend money on them you can make a suitable substitute from Phillips screwdriver bits. You just need to grind the tip of the Phillips bit a little then test the bit in a screw then grind a little more as necessary until the bit fits the screw properly. When properly ground the bit should be able to easily stay in the screw without help. If you need more explanation, just ask - pictures are available.
Good luck with your "new" bike, and be sure to ask any questions here as there are lots of knowledgeable folks on this site. Ed