The best choice for grasses would be fine fescues, which includes creeping red, chewings, hard, and sheep fescue. Mow around three inches or so. Fertilize less than for lawns in full-sun areas, with an annual nitrogen rate of about one to two pounds per 1,000 square feet. Try to limit traffic over the lawn. Even when following these practices, ground ivy may still appear.
Hand pulling is one control option.
Herbicides are also available to reduce the population. Check product labels for these active ingredients. Three Way Lawn Weed Killer, etc. Read and follow all label directions. Herbicides containing 2,4-DP or triclopyr may also be effective.