Couple of ways, but both are identical in how they work.

First step is to NUKE everything you want gone with Round Up Pro and wait a week for it to work.

Next you have two options of materials of being either Landscape cloth, or my favorite is cardboard boxes.

Once the bed is lined with either Landscape Cloth or Cardboard, pile on the mulch

Your only option is to dig all of the bermuda up. The plastic sheeting as likely not killed all of it, especially since it is so early in the season. I made flower beds where Bermuda used to be and have had great success. I took a sharp edge cutter and cut 1 ft squares and then shoveled them out. Once all the burmuda was gone (with about 4 inches of soil) I brought in a bulk delivery of rich mix soil amendment and mixed it with existing soil with a rear tine tiller. This is the proper way to make a bed. Your plants will grow faster and healthier and u won't be fighting bermuda popping up all season. You are also going to need to put some sort of edging around the entire bed (including the part bordered by your fence) because u and your neighbors grass will creep into the bed. Do the right work now and enjoy your bed for years to come!