Non Uniformity in MFCC Training Data
Definitely don't use the whole file as a single frame. 
As Alexander says, you want to use 25 millisecond frames with a 10 second overlap/hop (the vocal tract is stationary over this duration, so this is a common choice in speech recognition).
Each frame of audio makes a certain number of MFCCs (say 20). This number is constant for each frame of audio. But a file of different duration will, of course, have a different number of frames (and thus MFCCs). The size of your features (20) should always be uniform, but the number of observations (how many frames) you get from each file usually will not be, and that is okay.
Once you have extracted these acoustic-features, you need a machine learning algorithm that can "learn" what MFCC features are characteristic to a particular speaker. This is a very broad topic, but I can point to you this project: https://github.com/ppwwyyxx/speaker-recognition where you can see some of the machine learning algorithms used.


Good luck!