You are talking about the one from WTK book 1, right? With all of the arpeggios?
Will the circle of fifths be of any help? Kind of. Strong chord progressions often include sequences that move by fifths. You can find many places in this piece where chords move by fifths.
At the start of this piece, Bach establishes the key as C major by starting with a C major chord and going through a very straightforward sequence that leads right back to the C major chord. Today we would think of this as a ii-V-I progression. The second measure is an inverted D minor 7 chord, the third measure an inverted G7 chord, and then you have the C major chord in root position. If you look at your circle of fifths you will see that going from D to G is moving down a fifth, and going from G to C is moving down a fifth. Very common idea that recurs in this piece and most tonal music.
Why are there so many accidentals if it is in C major? Most of them are secondary dominants. In measure 6, we have a D7 chord. It is different from the Dm7 chord in measure 2, because the F has been raised to F#. Why? Well the next chord is going to be G major, which is V in the key of C. But in the key of G, the V is D (or D7 in this case). So the F# sharp here is coming from the key of G major. So measures 5-8 in total, proceeds Am - D7 - G - C maj 7. If you look at circle of fifths you can again that we are following around it. This kind of like, a little flavor of G major, but we aren't actually going to stay in that key for the long run. Bach will mess with our expectations for a while and then eventually reestablish the key as clearly being C major.
In other cases, the accidentals are in the form of a diminished seventh chord. This is kind of another type of secondary dominant. The diminished seventh chord is like the vii chord of some key, which usually wants to resolve up. So for example he has a C# diminished seventh chord which leads to D minor (which eventually brings us back to C major).
Topic: Bach Prelude in C major