The idea of mixture harmonies is a simple one: it is simply chords borrowed from the parallel key, hence these are sometimes referred to as borrowed harmonies/chords. In general, it is more common for major key to borrow from the minor then the other way round.
The first step towards learning these chromatic chords is therefore to be very familiar with the diatonic chords for both the major and minor keys so that you know what to borrow (or can recognize them when analyzing). Review this if necessary.
You'll now appreciate the importance of differentiating the chord types with your Roman numerals. For example, for the progression I-I6-iv-V7-bVI, we will infer that it starts in the major key but quickly introduces the mixture chords iv and bVI, the latter creating a kind of borrowed/mixture interrupted cadence. Usually, once a mixture chord is introduced, the progression does not quickly revert back. So, in this case, should we wish to expand it by inserting a supertonic chord, it will be I-I6-iv-iio6-V7-bVI rather than I-I6-iv-ii6-V7-bVI.
There are different kinds of mixtures composers have used, the classification depends on the nature of the borrowing and change.
When it comes to functional analysis, note that borrowed chords typically substitute for their diatonic counterparts, hence share the same harmonic functions:
e.g. I - I6 - IV - ii6 - V7 - vi (diatonic)
I - I6 - iv - iio6 -V7 - bVI (with mixtures)
T PD D (T)
Needless to say, mixture chords can be tonicized too, as long as they are major or minor chords.
Once you've grasped the basic idea here, explore how composers in the past have exploited the expressive potential of mixture harmonies--whether it's merely engaging in the aural pleasures of playing with harmonic colours or conjuring powerful emotional portrayals through harmonic shapings that take advantage of borrowed harmonies.
Neapolitan chords are essentially a sub-category of mixture harmonies. Here, instead of borrowing from the parallel major or minor, we borrow from the parallel Phrygian mode to obtain a bII chord.
The most common form is the Neapolitan sixth, which is the chord in its first inversion. These usually function as the predominant.
However, Neapolitan chords can occur in root position too, or even in second inversions. They can of course be tonicized.