I'm 14 & I'm a girl. I weigh 62lbs and my height is 5'1? Is that normal?
What's the average weight for a 14 year old girl?

62lbs? That's under 30kg... right? It sounds as if you're underweight.
Check out this link: Children First for Health - Teens - Health - Healthy eating - Your weight and BMI - Find your BMI (http://www.childrenfirst.nhs.uk/teens/health/healthy_eating/weight_bmi/bmi.html)
And try eating and exercising more.

No, it's around 31kg if I'm not mistaken.
Ow, you won't believe how much I eat! I get hungry easily. So, I eat a LOT! They say I have fast metabolism.

Yes you have fast metabolism. I ate tons of food and was 60 pounds all through middle school. I would eat a whole half gallon of ice cream just to try and gain 10 pounds.
Now I wish I could lose 10 pounds.

As long as you are healthy and not losing you should be all right.

Oh my goodness, I think that's very under weight. You're two inches smaller than me and you are the same weight as my 8 year-old brother! Oh wow. I weigh 97 pounds and I'm still under weight and I'm almost 14.