Discuss the idea of skipping a grade with your child first. He should always be in on every conversation concerning his education as the decision will ultimately affect him the most. Ensure that he understands every aspect of skipping a grade, including new classmates, being the youngest, and the possibility that he may not be the best in his class once he is learning with older children.
Assess your child's academic ability. By having your child take a series of tests found in books such as the Iowa Acceleration Scale: A Guide for Whole-Grade Acceleration K-8, you can present your case with paper proof that your child meets or exceeds the educational qualifications to move ahead a grade.
Speak with the principle and guidance counselor at your child's school to find out if there are other ways to accelerate learning without skipping an entire grade level. Your child could potentially take one or two classes with a higher grade but remain with her classmates in other classes. The main point is to make sure that skipping an entire grade will be the best decision instead of another means of academic acceleration.
Meet with members of the school to convince them that your child should be approved to skip a grade. Not every school will allow grade skipping, and most will be hesitant to allow it. However if your child needs more stimulation than they are recieving, then the school's faculty should be accomodating and more than willing to do what it takes to help your child grow.