There is very little guitar instruction on the chromatic scale, this is because there is yet to be a universal method of playing the scale. As we learned in a previous lesson, the chromatic scale is the name given to the scale that contains all 12 notes of the western octave. It is a great scale to base exercises on aimed at developing fretting hand finger independence, and also as a great scale to use in improvising in certain contexts; so in short, a great scale to look at if you want to learn acoustic guitar.
Chrom- is the prefix from the Greek word “chroma”, this translates as ‘colour’, therefore the chromatic scale means the scale of colour; it is given this name because the scale contains every note in the octave, every colour in the musical palette.
This scale really began to be exploited by composers such as Chopin in the romantic era of western classical music, so take the time to check out some of his compositions, they were usually very expressive pieces of music, but since their time of composition they have been used by students of the piano to develop complete control and virtuoso technique. The same can be applied when you learn to play the acoustic guitar.
The reason this scale is often used for fingering exercises is that all the fingering I have seen for the scale utilise all 4 fingers of the fretting hand so thorough practice of the scale will result in a high level of dexterity on the left hand fingers.
Here is the chromatic scale demonstrated on a single string, there are some tricky position shifts involved when playing the scale like this

It is more common to see the scale played in this fashion, again, the downside is that as you ascend the scale you descend the fretboard. This also doubles as an effective alternate picking workout.

This is another fingering I have seen for the chromatic scale. Although this one stays in position its downside is that it has 5 notes on each string so will always include a position shift on one of the fingers. The other downside is that the stroke you start each string with alternates with each new string.
This exercise doesn’t contain all 12 notes of the chromatic scale but it is rooted heavily in the chromatic context we have been looking at in this guitar instruction. Once you have ascended the following exercise you can move up a fret and descend the exercise. When you reach the bottom you can move up a fret and start over again. Play this slowly and focus on the strict alternate picking required to execute the exercise.