Creeping Charlie is a common name given to several plants. The most common one is a relative of mint, that has square stems and a strong aroma. It is a shade lover, exposing it to more sun would probably kill it.
Sheet mulching smothers out some plants that spread from rhizomes and runners, might be worth trying (put "sheet mulch" into the search here on Permies, lots of threads about it.) Put "Johnson grass" into the search bar here too, for ideas, it's a rhizome and seed thrower plant that is also an utter PITA to eliminate.
Might try taking it down to ground level with a mower set so low it's throwing dirt, with a bag on it (throwing the cut parts is spreading it more) many times, very often, keeping the plant from having any chance to recover. That will leave a bare dirt spot every time you cut, and it will erode until you can reseed it, but you can't reseed until the Charlie is dead, so you will probably lose soil to erosion.
It's also throwing seeds, so if you want it out, you will need to keep it from setting seed too.
Crowding it out with things you like more may work. Personally, I'd go for a plant that makes a dense root mat, like monkey grass (which is not a
lawn grass) that might take up all the space the Charlie is exploiting. Most opportunistic plants like this tend grow best on disturbed soils, with no competing growth, but not thrive if there's competition. (look up Pioneer Species, I'd put this in that category.)
The most important question is why do you want it gone? It tends to grow in shaded areas as a ground cover, if it's the same species I have, I like it, keeps the dirt from exposure and erosion where most things won't grow. The
bees adore it. Is it really worth the fight to eradicate it? Saying "I have a shade tolerant, sweet smelling, bee attracting ground cover!" might sit easier in your head than "I have a noxious weed." It's certainly less effort! :)
And I'll happily trade you Johnson Grass for it, I learned most of what I just wrote by learning the hard way when I moved that in this area any disturbed soil turns into Johnson Grass, and that stuff will get 12 feet tall behind my barn when I am not looking, I can't even get a brushcutter through it. Whee. I've learned to seed every piece of soil I touch with at least clover, it germinates faster than JG.