Purchasing an existing LLC
To purchase an existing LLC you need to have a purchase contract drafted and signed. This is typically done with the help of an attorney.
As far as business filing goes, in those few states where Articles of Organization list members the current managing member of the business would need to file a Certificate of Amendment with the state, removing the current owners and adding you and whatever members you want to be listed on the Articles of Organization.
Once the amendment is filed with the state, you would have to amend the EIN account with the IRS to change the name and social security number on record for the business, followed by an amendment of the company's Operating Agreement to include the new ownership and provisions of the business.
Beyond that, there may be some little changes to be made, such as the name on the company bank account and such.