I have a screening interview in a couple of weeks. I am confident I will be invited to an on-site after that. When do I tell my current boss I am in the process?
I don't know your relationship with your boss, but I personally would have an offer in hand before disclosing any information.
The human condition is always a factor, and resentment or a change in demeanor/culture can ensue if it doesn't work out with the new firm you are entertaining. I always give companies that I leave a 3-4 week notice, though. That gives them a better chance to figure out coverage or replacement, and also gives them opportunities for counter-offers should they not want to lose you.
The absolute best way to maneuver is to follow your own convictions. At the end of the day, you are the one having to navigate it all and you understand all the dynamics at play.