According to page 162 in the PHB, a strength 10 character has the following load limits:
Light: 0-33lbs.
Medium: 34-66lbs.
Heavy/Maximum Load: 67-100lbs.
Lift: 100-200lbs. (You can lift up to twice your maximum load and stagger around with it (5ft. per round).
Drag/Push: 500lbs. (5 times your max load, possibly up to 1,000lbs.)
A Cart costs 15gp and weighs 200lbs. As you don't seem to have access to the first level spell, Mount, you'll have to drag it as a person. I would certainly call a cart with wheels a "favourable condition", so you could load the cart down with as much as it can hold (as long as the weight of the cart and the weight of the gear don't total more than you can drag).
Now, it gets a little fuzzy rules-wise, but here are my thoughts. A cart is made of wood, with some metal (or it would weight quite a bit more). In order to reduce the weight of the cart, you'd want to make the wooden parts out of Darkwood, and the metal parts out of Mithril. That cuts your weight in half, to a 100lb. cart. The value of the cart would increase by however much Darkwood increases an item's price. Then, you load the cart with what you can. Personally, I wouldn't allow the load to weigh more than a ton, as I doubt even the Darkwood and Mithril could take the strain. The other consideration is that this is a small, two-wheeled cart. I'd figure it won't have a great deal of holding space, so make sure that whatever you transport is dense, to conserve on space. I'd guess that the small, two-wheeled cart wouldn't be more than 4 feet wide by 5 feet long.
You could try the process with a wagon, which while it would certainly hold more cargo, it is much heavier (it is designed to be pulled by a team of at least two horses).