Oh, I hate purslane! About all you can do is pull it before it flowers and goes to seed. Killing the lawn and replacing it again won't get rid of the purslane. Roundup only gives it a stomach ache as the waxiness of the leaves causes the Roundup to slide right off.
Some will tell you its a good vegetable - I will disagree with them.
And even good vegetables have their consumption limits (unless you can convince your friends they like it too).
Daisy