I was unable to bring up the model number but for a Kenmore dishwasher you do not remove the door.
On a Kenmore dishwasher the most likely part to fail is not the metal door spring but a rope that connects the spring to the door. To check this first remove the access panel. See 1st image below.
The dishwasher will most likely have to be moved out about 12 inches to get good access and replace the rope but the 2nd image shows what you're looking for.
The spring is part key number 18 and is metal so it's not likely to have broken. The rope is part key number 24, part number 8194001, and has been known to break. The rope is routed over part key number 9 and attached to the door. With the lower access panel removed you can reach in and feel for the spring and rope link.
To remove the dishwasher from the hole with the door open remove the two screws that are mounted into the counter top. With the screws removed it should be possible to pull the dishwasher out of the hole between the cabinets.
How far you can pull it out will depend on how much slack you have in the water line, the drain hose and the power connection.
Be sure power to the dishwasher is turned off while servicing.
I hope this information is helpful resolving your problem but if you have any additional questions then submit another question with additional details for further assistance.