Toilet backs up into sink and runs

toilet backs up into sink and runs
You have two complaints. Let's take them one at a time.
You have a blockage downstream from the sink. That's causing the back up.
The solution would be to rent a sewer snake,(see image) and snake from the kitchen roof vent. Put out enough cable to reach the base and 20 feet more to reach the main.
Please explain what you mean by " it runs"? Does it overfill the tank? Does it leak out of the tank when it's filled? Before In can help I have to know what it's doing. Back to you, Tom