Don't run. Knee pain like you are describing is not something that happens as a result of training. It sounds like a real acute injury.
When you are healthy enough to run again:
3 Problems:
1. Poor form
2. Too much weight
3. Muscles haven't adapted yet
Poor form: You need to get fit enough so that you can maintain decently good form throughout your run. If you can barely make it through a run and you are dogging it at the end, then your form has broken down well before that point. If you can maintain good form, then you can carry a little extra weight and be fine, but if you have poor form then any extra weight is going to multiply force and impact of every foot strike.
The other two are self explanatory. All three will improve if you run more, but you can't run more unless you fix numbers 1 and 2 or you will get injured.
A. Do some other aerobic cross training like riding a bike or swimming. This will improve your aerobic fitness and help lose some weight.
B. Do not do runs that are longer than you can maintain good form for. Do most of your runs at an easy pace. If your goal is to run a 5k at 9:00/mile pace, then an easy run for you might be 3 miles at a nearly walking pace. That's just how it is until you are more fit. Don't try to be a hero and pretend that you should be running faster.
C. Don't overdo the running. If you can only run 10-15 miles per week now before you start feeling problems, then that's your limit. You can only increase this slowly over time as your body adapts. If you push your limit too far too fast, then you will get injured.
D. You can run fast sometimes, but for now you should only try it once or twice a week. I don't know how well things like the McMillan calculator translate up to these paces, but the tempo pace for your 5k goal would be 9:14-9:32. You could probably put in a tough effort of about 2 miles in that pace range once or twice a week.