The only sure way is if your system uses an MXM-standard video card. MXM ensures easy interoperability of different video cards, but only some systems use it. To check if your system uses an MXM-compatible video card, you can check the card against these specifications
(link is a pdf). If the card matches, all you have to do is swap out the old card and swap in an existing MXM-compatible card.
If your video card is not
MXM-compatible, then the process is a bit more difficult. You essentially will need the video card from an updated version of the exact same computer model you own. This is only possible if your system maker typically uses the same chassis for several different product generation revisions.
If your video card is not MXM-compatible and your system isn't part of a recurring series, then it's nearly impossible, as you then have to go through an extensive process of trial and error in attempts to find something that will both physically fit in your system and work when you get it in there. This process will likely cost you the same as a new system as you try out (and potentially ruin) different video cards, so it's not recommended.
The specific things that need to match are: physical connector, physical screw holes, physical size and dimensions of card, power draw (equal or less than original), thermal dissipation (equal or less than original), and often brand (you'll only be able to replace an AMD card with an Nvidia card or vice versa if you've got an MXM slot).