To control weeds around the strawberries... well. It just gets tilled over once year and I manually go out there with gloves and pull it out when it comes back. Same with weeds between all the strawberries. I manually pull them all out. I don't use chemical sprays or have any mulch. My strawberry patch is only about 3 feet wide and 18 feet long though and I plant things too tightly to reduce empty spots where weeds can grow. The rest of the area around the berries grows beans, peas, tomatoes, rhubarb, and potatoes.
Then again though. I am also the person who takes a shovel and manually turns over the soil if my lazy little brother doesn't get the tiller out. Takes a few days but it gets done!