Hi Brent!   I love periwinkle (also known as Vinca minor or Myrtle) and have a nice patch underneath a large maple tree.  I find that once you have a thick enough base of periwinkle, it tends to keep other weeds at bay. That doesn't mean that I don't have to pull the occasional weed out of my patch -- but as long as I spot the rare intruders and pull them before they flower/can spread, my myrtle patch doesn't require much care. 

From what I can see in your photos:  you have a bigger job to start with here!  I can see myrtle interwoven with weeds in the foreground of your photos.....but the further out the image goes, the harder it is for me to differentiate the plants on the hill.  So I'm making a few assumptions here:

1. The mix of myrtle and weeds that I see in the foreground of your photos continues on back (i.e.: you have myrtle you'd like to save throughout the hillside section all the way to the trampoline)
2. You're not asking about the mix of grass and moss on the lower-left-hand side of the photos (appears to be under a tree and where another tree might recently have been removed).
3. I don't think I see any invasives in the mix (thank goodness!) -- but it's still wise to wear gloves and cover up before wading in here.

Based on those assumptions:  

• To avoid sacrificing the myrtle: you'll need to hand-pull the other weeds.  We recommend cultural methods of control before resorting to chemicals and harsher methods;  but weed killers wouldn't be a good option for you regardless:  periwinkle is a broadleaf plant, as are the weeds I see....so applying a weed killer would be fatal to the periwinkle, too.  If it were me:  I'd take this on in smaller bites:  one section at a time.  It's a lot of work at the start.....and the weeds that have been growing on the hillside have probably left seeds (and there may be roots left behind after weeding) that will try to perpetuate their species.  But if you are persistent, you can win this battle.
• Once you clear a section, I recommend adding more myrtle in to fill open spaces.  You could, of course, take the long path and wait for the remaining periwinkle to slowly expand and fill in the gaps. You'd probably end up pulling more weeds, but could eventually have a full hillside of periwinkle that doesn't require a lot of maintenance.  If you have another ready source for periwinkle:  tuck plants into the gaps....help them get established (water as needed in the first season).  I give my periwinkle a little organic slow release fertilizer each spring.  
• If you hope to clear the grass/moss area OR if you have sections on the hillside without periwinkle plants, you might find that a "loop hoe" is an efficient way to remove weeds.  (You could consider this approach, too, if you decide to cut back on the stretch of periwinkle.)
• Finally:  soil is one of the most important factors for plants.  Periwinkle prefers a more acidic pH, but is very tolerant of different soils (sand, clay and loam can all work). I would normally recommend obtaining a soil sample and amending your soil, if need be, to create the ideal environment for periwinkle;  but this plant is so adaptable that it is considered invasive in some states (not here in Vermont) -- so if you find that your periwinkle's doing just fine without amending the soil:  you probably don't need to do so.  

I'll end with a short exerpt from a University of Missouri Extension fact sheet on ground covers (I like Missouri's documentation because they have similar climate zones to Vermont's).  

Periwinkle, Myrtle or Vinca (Vinca minor)

Periwinkle or myrtle is an excellent evergreen ground cover with dark green foliage and purple, blue or white flowers depending on the variety. It blooms in April and sometimes again in the fall. The plant grows about 6 inches tall, spreading in all directions by sending out long trailing and rooting shoots which make new plants.

It likes shade but will grow satisfactorily in full sun. The foliage color is richer green in partial shade, but more flowers are produced in the sun.

This ground cover is most commonly used for underplanting trees and shrubs, on shaded slopes or on the north side of buildings.

Rooted cuttings or established plants are normally spaced from 12 to 18 inches apart. At a 6-inch spacing, a complete cover will be produced in one year. Plant in the spring in areas with severe winters.

Spring-flowering bulbs interplanted with periwinkle will lend color and interest to the ground cover planting. Daffodils are particularly well suited because they bloom with vinca and do not require frequent division.

I hope this helps, Brent, and that you will have a hillside full of thriving myrtle by this time next year.  

-- Robilee Smith