Hi Brent-
It really depends on what you are going to replace the weeds with. There are many low growing and low light requiring ground covers and plants to choose from.  One idea; If you want the periwinkle to take over, consider first testing the soil, then amending as recommended (most likely with fertilizer and lime) then (if wanting to avoid herbicides) covering the soil with cardboard and topping with a few inches of a fine bark mulch or compost. Next, transplant periwinkle plants from your other thicker areas by digging out and putting in holes cut through the cardboard for them to take hold in the new area.  This can work for other ground cover plants as well, not just the periwinkle.  After a few years, if the ground cover is well maintained (fertilized and weeded) , then it should fill in before the weeds reestablish.
If you want to go with herbicides, it will depend on the types of weeds present, but a general, broad leaf herbicide will take care of most species (except grasses). This would reduce the need for the cardboard and allow you to lay the mulch directly on the soil, then proceed as otherwise recommended above.
Please get a hold of me directly for more information as needed.