CH- Welcome to Motocycle Mechanics 101! If you have a local community college, they MIGHT have a basic small engine or perhaps even a motorcycle mechanics course. If they do, sign up! In any case, Loud was correct on this one! Get a factory service manual (Kawasaki) for your bike year/model as well as a Clymers. Scan through the sections from front to back and you will have a better sense of bike systems and how they inter-relate.
In this case, you are mixing the intake (fuel) system with the electrical. While these systems depend on each others' correct operation to allow the bike to run optimally, they are not directly related.
A motorcycle needs electricity while running to operate lighting systems, horn and the coils/ignition. In essense, only the coils and ignition need to be working for the bike to be rideable although lights/horn can be considered an essential item due to safety concerns. Snce your bike died and it appears if I read this right, the new battery allowed it to run but then it died again, the battery would appear to be not charging. The way the bike's electrical system works, the alternator produces AC current and sends it to the regulator/rectifier and Loud explained. The AC turns to DC in the proper voltage range here.
The likely culprit if the new battery isn't charging is the regulator/rectifier. It is a small 4x4" box about 1" deep located in the area somewhere near your battery since one of the outputs of this gizmo connect to the positive battery. The connection will be on the starter solenoid which is the relay that kicks your starter over. You can find it by tracing the positive battery lead to see where it connects. This is the starter solenoid and on the same post, a white or white/red wire with an eye connector that slips over the post under the battery cable will be connected. This white or white/red wire traced back will lead you to the regulator/rectifier (you will have to trace through ONE dual connector I think.
Anyway... you don't really have to find the reg/rec for this test. In order for the battery to charge correctly, the reg/rec MUST do its job properly. It MUST supply a positive DC voltage that is somewhat greater than the 12VDC the battery is rated at. This extra charge is what causes the battery to be charged and keep charging. When the bike is idling, the voltage from your reg/rec should be about 12.5 Volts DC or VDC. At 4000 rpm, the voltage should be about 14.5 VDC. If it is less or more, the regulator/rectifer is BAD or the connections to the reg/rec are dirty or are bad. To check voltages, get a multimeter, put it in VDC scale and place the RED or positive lead on the positive battery terminal marked with a PLUS + and the black or NEGATIVE multimeter lead on the negative battery terminal marked with a MINUM -. With the bike running at idle, note the voltage then rev it up to 4k rpm and recheck and report back in with the readings. If they are off what I noted, the reg/rec is likely bad.
OK... if you find good voltages as noted, the problem will be either with your alternator (new one needed) or the wiring. The wiring can be frayed or have bad connections. You can easily check the connections if you know where the reg/rec is located... open up the connectors into and out of the reg/rec and clean with contact cleaner and coat with dielectic grease. The last possible problem area that would cause the bike to not charge is the ground. This is the BIG wire coming off your negative battery post. Follow it to where it connects with the engine. Make sure the point of connection is CLEAN (not corroded) and tight at both ends of the cable.