Are you using a serger? Also, are you really using fishing line? Or is it an invisible nylon or polyester thread?
I'm no expert, but I did some lettuce edges years ago on my serger. The fabric has to be a knit or else cut on the bias. This is because the edge must stretch in order to get the lettucing effect. As I recall, I set the machine up for a satin stitch (narrow stitch that is close together) and set the differential feed to stretch the fabric the maximum amount.
You don't need invisible thread to do a lettuce edge. Any type of thread will do it. Are you intending to use fishing line inside the stitching? I never did it that way, but I think the trick would be to feed the line in such a way that it is not cut by the blade but still enclosed within the stitches. I would think you need a special foot for that -- something similar to a pintucking foot with a guide for the fishing line to be fed in. For a serger, that might be the foot used for adding pearls-on-a-string to a garment.