Step 1: Separate Key
The first thing to do is to start considering Dorian as a separate key, just like you would any minor key. For example, we don't think G major when asked to play in E minor, even though E minor is the 6th degree of G major. So, why would we, therefore, think G major when playing A Dorian. Yes, this is the basic theory (which you can find all about here), but when it comes to playing practically on the guitar, this doesn't help!
Step 2: Minor Mode
Dorian is a minor mode, simple. So when it comes to playing it on the guitar, to get the most from the mode, we need to compare it to the natural minor. Here are the two scales side by side:
A Minor: A B C D E F G A
A Dorian: A B C D E F# G A
As you can see, there is only one note difference between the two. This is the 6th note. In the natural minor it is flattened and in the Dorian, it is a natural 6th. This is, therefore, the note we should be targeting to really pull out the essence of that Dorian sound.
Step 3: Dorian Box
To get this nice and clear on the guitar, we will take one position, based around the Pentatonic box one in A Minor. Below you can see three scale shapes and how they relate to each other. The Pentatonic first, then the A minor, then the A Dorian. Learn this Dorian shape and you'll be away!