You're 4kg out from the "average" (and, btw, WHO said 103lb is the average weight of kids your height and age? That certainly doesn't sound average to me, since the average kid is following in their average fat parents' footsteps, to be brutally honest). Do you know what that is? Insignificant! Stop comparing yourself to other kids. Stop placing so much value in this perception you have of what your body is supposed to be like. The difference is insignificant. 114lb is not fat, and it's far from any risk of becoming overweight.
Now, I'm not saying this to discourage you from being physically active and eating good, wholesome food. I highly encourage this. But I honestly think you're seeing a problem that isn't there. Teenagers are taught to judge themselves negatively, cynically and unrealistically. You're weight is perfectly normal. Your height:weight ratio is exactly as it should be. Anyone telling you otherwise is lying.
Now, there's a lot of information around this site about what you should be doing with your body. Good nutrition, good physical work (generally expressed as exercise, sports, games, manual labour or training), and good rest. This doesn't have to be running, and I really wouldn't recommend daily running to anyone who isn't an experienced runner with a sports or career-specific reason to do it. Of course, if you really do want to run, then run, but if you say you want to do it because you've been taught that it's an arbitrary requirement, then please reconsider and think about what physical activities you want to do - I'm fit, lean, and haven't ran more than about 100 metres in nearly 3 years other than when I've actually had to run.
Whatever physical activity you do, it should last 20-60min, and you should do at least 3 days of activities each week, preferably more. The duration is probably more important at this point in time than what distance you cover, although if you do want to run, 1mile slow and steady would be a good starting point. Activities could include walking, running, cycling, swimming, rowing, dancing, any number of team or solo sports (netball, football, hockey, gymnastics, martial arts, strongman, weightlifting, bowling, etc), speed training, agility training, sled dragging, strength training, circuit training, gardening, walking/playing with pets, and the list goes on.
You mentioned that you would like to see an improvement in your muscles. Shockingly, muscle mass is in fact mass, so this is yet another reason to abandon your concerns about your weight. If you want some muscle tone, then activities that you cannot perform for more than 1min without resting are the way to go. Strength training on a balanced program is probably the most reliable way to go about this. You'll only need to do 2-3 fullbody strength training sessions/week. Look in the weight training section for more information.
