Your budgie doesn't look dirty at all in that picture, I definitely don't think he needs to be washed, I agree with Ana I suggested Dawn dish washing liquid previously as I thought your budgie had some sort of dirt stuck to him before seeing his picture. As I said also my budgies all go this colour when wet. Budgies need their feathers to be oily and they are the cleanest birds around.
I think what the problem is the new feathers are pristine much like a newly painted wall, over time daily grime and dust will have some effect on the colour.
Your budgie looks absolutely normal .