You have two ways to go.
You can keep your existing DirecTV receiver, buy an older "Series 2" standalone TiVo (not the "Dual Tuner" model and not anything newer). The TiVo would go between your DirecTV receiver and your TV. You would be able to record only one program at a time, in standard definition, and could not watch "live" while another show records. You would have to pay a TiVo service fee to TiVo (not sure what it is nowadays - $14/mo or so perhaps?)
The other is to replace the DirecTV box with a box that combines a dual-tuner DirecTV receiver with a DVR. DirecTV offers three kinds: standard definition (with a non-TiVo interface), high-definition (non-TiVo) and high-definition with TiVo. For all of these you would pay a DVR fee to DirecTV (currently $8/mo). For the TiVo box (model THR22), you would pay an additional $5 TiVo fee to DirecTV. This would allow you to record two shows at once, or record one and watch another live. This option would also add some services such as scheduling over the web, downloadable PPV and more. The non-TiVo boxes have additional networking features. You would probably find that DirecTV would provide one of these boxes at little or no cost, including free installation. HD would require a new dish.
You could also buy a used, older DirecTV TiVo box. It would have to be one that has a "Receiver ID (RID)" and would be standard-def only. This would get you out of the $5 THR22 TiVo fee. But be careful when buying a used box - many of them are from accounts that had balances owed, and DirecTV won't activate them. They also won't activate a "non-RID" box that wasn't previously on your account.
I hope this helps.