Matthew,
Welcome aboard. the good news is that your bike is so simple that we will be able to get to the bottom of this easily. The bad news is that it is going to take some patience. Others may give you other good advice but here is mine.
1. You have to start at the very beginning. I would suggest that you consider a new set of points and condenser, probably a new plug, an investment in a Top Dead Center tool, a volt meter (even a very cheap one is fine - I much prefer the analog type) and last but not least Pete Snidals manual - it is particularly good for older models.All can be found here
http://www.royalenfieldusa.com/dead-center-findertiming-tool-p-469.htmlhttp://www.royalenfieldusa.com/original-bullet-motorcycle-tuneup-save-more-than-p-495.htmlhttp://www.royalenfieldusa.com/snidal-royal-enfield-motorcycle-repair-manual-edition-p-242.html2. Your ammeter should swing to the left every time the points close which is most of the time. It indicates that current is flowing through the points, hence the discharge on the meter. If it is acting erratically it may be that the points are worn out or (more likely dirty). If you don't want to spring for a new set, at least clean them and gap them at .015"
3. Next check your timing (highly likely that it is off). The Top Dead Center tool is invaluable for this, but in a pinch I have done it with a pencil upon which I made marks 1/32" apart. Long story short, using the tool of choice, bring the engine to TDC of the compression stroke (not the top of the exhaust stroke), Then back it down enough to take the slack out of the timing gears and move the piston up to 1/32" before the spot you marked at TDC. With the piston in this position, loosen the backing plate of that the points are on and move the points plate until you either hear the spark plug "snap" or see the ammeter go from deflected to the middle. Then tighten everything up.
4. Everyone ignores this, but check your battery. A good battery will have at LEAST 13 volts in it. A low battery will mimic all sorts of things including the symptoms you describe.
Then follow the starting procedure. Use the choke, and I like the idea of kicking it through a few times before turning the key on and trying to start it.
5. When you kick it, put your weight into it. Strength is not as important as technique. A long kick with follow through is very helpful. I like to hit the top of my boot on the underside of the foot peg. They almost always start in the last 1/3 of the kick.
A couple of things to look for on this bike
1. The alternator won't charge unless the key is in the second "on" position. It will run in the first, but not charge.
When your bike runs the ammeter should be three-fourths of the way up the scale. If it hovers in the middle it is not charging and will need attention.
2. When you have the distributor cap off, move th points cam back and forth with your thumb and forefinger. It should move back and forth freely. (rotate a few degrees back and forth - I do not mean rock back and forth). It not check and see if the washer under the bolt that holds the cam on is keeping it from turning freely.
3. At some point (like when you are replacing the points) take the points cover off and inspect the back side. In some of the older bikes the advance weights would hit the back of the condenser.
This should be enough to keep you out of trouble for a while.