If you can't run 10k without stopping, then you are either running too far, or too fast for your current fitness.
I would recommend a program created by a cross country coach named Barry Pollack, dubbed the 3:2:1 program. In this, you have 6 runs per week, 3 short, 2 medium and one long. Your medium run is double your short runs, and your long run is 3x your short run.
To get this, go out and run at a slower pace than you normally do, and see how long you can run. Mark the distance where you have to stop, and that will be your long run. However long that took you, that is the time for your longest run. If that took you 20 minutes, then your 3 short runs will be 7 minutes each (Approximately) and your 2 medium runs will be 14 minutes each.
Even though that doesn't sound like much, it's a good starting point and steadily increasing over time will give you great benefits. Your goal should be to add about 5% each week. So, if your long run this week is 20 minutes, next week it should be 21 minutes, and so on. Remember to add time to your other runs as well as your long run increases.
Don't worry about how fast you are running to start. Most people run too hard on their easy days, and not hard enough on their hard days. I suspect that you are trying to run too fast, which is why you always have to stop. Slow down, and make your goal completing the full distance, not how fast you can do it. Speed in endurance running comes from putting in consistent training day after day. Your times will drop naturally, and once you get to the point where you can run a full 10k without stopping, then you can look at adding extra speed work if necessary.