Is 600 mg of vancomycin every 6 hours appropriate for an 11 year old weighing 42 kg then increased to 750 mg every 6 hours then 1250 mg every 6 hrs?

No: 5000mg of vancomycin per day is usually way too much for someone weighing 42kg. Blood levels can help to determine the correct dose, and I suppose under unusual circumstances, that high a dose might be required. But I would ask about the drug levels and why such a high dose is being used.

Vancomycin: The normal dosing for vncomycin is15-20 mg/6hours. The 600and 750 mg dosing is ok. The 1250 mg dosing appears to be too much