How to make salt in burger permeate throughout the entire meat?
This 'trick' is used by many a fine chef. Meat is salted, wrapped and refrigerated hours or even overnight, prior to cooking. Yes, salt will draw moisture from the meat initially, but over time that moisture will migrate back into the meat taking the salt with it! The salt also breaks down the protein structure in the muscle rendering a more tender steak.
The steak is rinsed and dried (a wet steak will steam not char), seasoned, and fired.
Ground beef can be handled in the same fashion. Using a medium or coarse salt and erring on the less salty side, you can find the right proportion to suit your taste. Season before forming or after. Some contend that the seasoning is made bitter by the heat and flame of the grill. Go figure.

Other tips: The meat should be mixed and formed with a gentle hand — use only as much pressure as it takes to hold the patties together. For almost foolproof rare or medium-rare meat, go right from fridge to grill.