First, an anecdote. I was visiting a musician friend, and we were standing at his piano (he also had a harpsichord, of which I was quite jealous) and chatting about music and probably sketching out a few ideas on the keyboard. I happened to play a dominant seventh chord, followed by... nothing.... at which his wife (also a musician) rushed from the next room, shouting "resolve! resolve!" and immediately hit the corresponding tonic chord.
This impressed me in two ways. Firstly, as an exhibition of her pitch recognition (in being able to find the right tonic!) and also as a demonstration that tension and resolution in music is very subjective. To my ear, the dominant seventh chord was quite happy to stay where it was - but she, with her different tastes, needed it to move on.
On to your question:
Tension and Resolution is the play and movement of a chord progression.
I wouldn't want to define tension and resolution that way, because they can happen in in pieces of music that couldn't easily be described in terms of chord progressions.
When building tension within a chord we start with a key, let's say C Major and use multiple high ratio notes stemming from the chord root note depending on the level of dissonance you want to produce within that chord.
Starting off with a key is a common approach.
"Use multiple high ratio notes" isn't a common description of a compositional process, though I think I know what you mean from your other questions.
if our first C chord contains the notes C, F and B...
There's nothing yet inherent to the selection of notes C, F, and B that would tell me as the listener that the chord was 'rooted' on C, or that we were in C major. It's quite a dissonant sound in itself, due to the tritone interval between F and B.
The information I have gathered states that the successive chord would approach the tension in the latter chord on a note to note basis...
...we would create resolution in our successive chord by using E, G and A which on a note by note basis constitutes resolution of a chord because they happen to be 3rds of the previous chord. Is this the adequate process of building chord progressions through tension and resolution?
No, you can't just look at the voice leading between individual pairs of consecutive notes. Ultimately, the scale of the problem is that each note in a piece of music can be said to have a relationship with every other note; that's a very tricky problem to solve, which is why people tend to look to established methods, such as triadic harmony within the major-minor system, for example.
If you wanted to create resolution in a C major context, why not move your F down a semitone to E, and your B up a semitone to another C? this would be smooth voice leading that would give you two Cs, and an E which is the major third of C.
If not what are the most fundamental methods?
Every style of music ultimately has its own approaches. Common practice harmony is different to Jazz, which is related to (but still different from) Blues... and then European folk music has its own traditions, etc etc…
It takes a talented musician to master more than a few styles. Coming up with a general theory of tension and resolution in a way that would encompass all of those styles seems very difficult indeed.
From your other questions, I think you may be trying to come up with a general theory of harmony based on "the science". That sounds great - but you do need to be rigorous with your science! You can't, for example, assume that the level of consonance observed when notes are played simultaneously is necessarily the same as when the same notes are played successively; When you base notes of different ratios on the 'root' of a chord, you can't pay attention only to the ratios they have with the root (and not with each other).
