Hey Joe,
Congrats on your purchase! We usually go ahead and serve the 3/90 day notice to quit on the door, along with a letter similar to the one you mentioned, on the day we purchase the property. We also put on there a cash for keys option and that usually sparks a call. We only flip, so we try to work something out with the occupants so they move in a timely manner.
If they don't want to talk/negotiate with us, we just start the eviction after 3 days. There are some risks starting the eviction before you have recorded your trustee's deed, but we have never had an issue with it. I believe you have 10 days to record after you start the eviction, but you definitely need to speak with an eviction attorney about all that.
It's also important to make sure the occupants are not bona fide tenants. If they are, you will need to wait until the 90 day notice expired before you can start an eviction (assuming they continue to pay you rent)
Hope this helps...