Can I connect a dishwasher to a garbage disposal instead of the sink overflow?
You can hook up the dishwasher to the garbage disposal - that is a normal installation. There should be a location on the disposal - typically a plastic part that gets knocked out to connect the dishwasher hose. You will probably need an adapter for the disposal - this often comes with the disposal but is probably long gone. If you post the model of the disposal, we can provide more info.

"Drilling" sounds like you may be referring to a traditional air gap in the top of the sink. Dishwashers can be connected that way, but they can also be set up with a high hose loop. Basically the drain hose from the dishwasher loops up to just under the counter, effectively forming an air gap (the only way sewage could get backwards into the dishwasher would be if the sink filled up to the same height as the top of the hose loop). The other end can go into the disposal. If you do this then the old air gap can be removed and the hole used for water purification or soap dispenser or whatever you want. See  one of my other dishwasher answers for a diagram.