Holy cow.
Most 'weeping' cultivars of whatever species - whether cherry, mulberry, or whatever - are grafted on a 'standard', at some point 2,3,4 ft or more above ground level - or, they are staked up to whatever height the owner or propagator wants them to weep from, then allowed to weep - and the supports are kept in place until the trunk has attained sufficient lignification(woodiness) to support itself. Otherwise, they're prone to run along the ground like a vine; sort of like yours is doing.

I must not have done staking right. Is it too late to stake it with a stronger and bigger stake now in early summer. I tried to change the stakes last year, but the trunk seemed pretty strong already. I did not want to use force to try to straighten it up. I let it grow in its own way and hope it would grow upwards, like most perennials lift themselves upward when stalks grow taller. Unfortunately, the weeping cherry is not the case. It does not seem lifting itself up at all. I have to do something now, or it will look like a spider forever. Please advice me.......

The mounding up you are hoping for might take many years. You must take a firm stake and insert it near the base of the tree, checking first to see if you can find a place where it can be driven through mostly unoccupied soil (pull some back with your hands if you have to to find out) - you don't want to hit a larger root with it and push it down, bark it or break it off.

Select a suitably flexible branch near the center of the tree and tie it securely to the stake at several points, so that it is parallel to the stake. You must assume the plant will grow sideways again from the point where the stake ends and therefore use a stake tall enough to elevate the crown as high as desired - after the stake has been driven into the ground. So, if you want the tree to have a top that branches mostly at 6' tall, you need a stake that is more than 6' long.

I am not sure if I get this right. OK, if I select a suitably flexible branch and tie it to a stake, wouldn't I just drive this branch up merely but the rest of them still flopping down?

The staked branch becomes a trunk, with a new elevated top above it. Eventually the existing low branches are removed. Or, the tree is grown branching to the ground. It depends on what you want - you can do either.

Where did you get that tree? Usually they are on a "standard" trunk 4 - 7 feet above the ground...

You can train it as suggested, that will do the trick but take a few years.

I'm more curious where you got the tree!

I got it from Burgess. I came in a 1" pot.
So.....it's not going to straighten up right away, but I have to keep adjust the stakes from time to time, as long as straightening it as much as possible, right?

As the staked shoot grows taller, you tie it in at intervals to keep it growing up the stake - like a vine. Nothing should be happening to the stake itself after it is first installed. After a good trunk is developed then the stake and the ties can be removed.

It might take up to 5 years to get to where it can stand without a stake.

After I stake one leader up, do I have to cut its branches at the lower part?

Only if you want to get a bare stem with all the branches at the top, like a parasol or mushroom. Otherwise you could grow it as a mound of branches, like when a weeping laceleaf Japanese maple has been trained up a stake to grow taller. Trees pruned and trained or top-grafted in nurseries to have branches only near one another at the top of a bare stem are called standards. Those grown with branches distributed up and down the trunk (although often not all the way down) are called feathered.

Oh, they even have terms for those conditions! How interesting! nice to learn them