How do you purchase tax liens on a house and buy it after the purchase?

You don't.

You may buy tax properties or tax certificates ( depends on how your state does it) at tax auctions that your county tax office holds normally once a year.

In some states you will get a special deed the day of the sale and own the property that day. In other states you get tax certificates, that allow you to own the property after a number of years ( 1 to 3 years depending on state) during that time the privious owner has the right to pay you back the amount paid plus normally 10 percent interest plus any improvements you have done to the property.

But you are not "buying the lien" and you are not buying the house latter.

You can buy the tax certificate and latter redeem it but it all starts at the auction.

Fr Chuck seems to have the experience to address this issue.

I do not, and, In my opinion, this question is NOT a tax question and should not be posted here.