Theoretically, it should be perfectly possible. However, there are some practical considerations.
Extraction of components: I've never taken apart a DVD drive but there are lots of videos and guides on the internet saying it is possible. They can provide better information on how to do that bit.
Laser Safety: DVD-drives have relatively powerful lasers, particularly for DVD-writer/RW. A quick internet search says they are Class 3 lasers (They are normally safe as you cannot access them). This could easily blind you. I strongly recommend you look up and follow appropriate laser safety procedures.
General set-up: The simplest approach would be to make a simple Michelson interferometer. Using one mirror as the reference arm and attaching another to the surface you are measuring. You will probably want a photo-diode or similar to measure the output signal and possibly a beam expander, at least for visualization purposes.
Distance measurement: The simplest way to determine distance from an interferometer is via fringe counting. This requires that the distance changes in a continuous fashion. If you try and measure a step or other discontinuity you will loose the count. The response time of your photo-diode may also place a limit on the speed at which you can change the distance.
Coherence: The biggest problem I can see is the coherence length of the laser. Laser diodes have relatively short coherence length. Therefore your signal quality will degrade as the distance between reference and object arm changes. Therefore, try and make sure you reference arm is reasonably well matched with the object arm length initially. In any case, this will likely restrict the possible measurement range to something on the order of a few cm.