In Texas the rules for purchasing a HUD home are very specific and I am 99% sure that it is the same across the nation since HUD is national.

HUD can not take over the property until the owner signs away the house or it gets foreclosed, which legally takes away the person's ownership. At this point it goes to an attorney to make sure the property is vacant and the title is clear to sell. This can take a minimum of 21 days.

After that the property either gets listed with a real estate agent or goes on HUD's website to be sold as REO (which usually still involves an agent.) If this property is in the the stage involving anattorney then it can't be sold yet, but you can contact the lawyer and ask who and or what real estate company it will be sent to. Then contact that broker and ask when is the soonest you may submit an offer to purchase the property. The agent and or broker will guide you from there.

Pending your state foreclosure laws, it is possible to purchase the property prior to the house being sold back to HUD. 

One of the steps in many states in the foreclosure process is to actually "auction" the property on the steps of the county clerks office.  In many states, the attorney representing the lender that had the primary position, will offer the property for sale as part of the foreclosure process.  Anyone can bid on the property at this time, other than the current owner.  Typically, the lender is the only representative there to bid, and they will bid exactly what is the current outstanding principle balance.  The sale is usually posted at the clerk / recorders office with a date, time and outstanding lien balance.  You can bid at this time to purchase the property. 

You may be required to put money down at the time of the sale, or have the ability to pay the balance within a short period of time.  Terms will be posted, or a contact will be given, but it is typically cash as the time of sale.