How to put chords to a bassline?

Let me start out by saying that you can do anything you want with those 2 chords.
if you want to make it sound like it has a key center, then here's what you do:
the C to Eb bass movement implies a couple of keys.
The two keys that come to mind (that would be the most practicle) are Eb major and Bb major.
What's great about scales is that every note in the scale can be a chord in a key center.
if we were in the key of Eb: (Eb,F,G,Ab,Bb,C,D)

(scale #) (chord)(interval) (Chord) (notes)
the 1 chord = major (1,3,5) = Eb major (Eb,G,Bb)
the 2 chord = minor (1,b3,5) = F minor (F,Ab,C)
the 3 chord = minor (1,b3,5) = G minor (G,Bb,D)
the 4 chord = major (1,3,5) = Ab major (Ab,C,Eb)
the 5 chord = major (1,3,5) = Bb major (Bb,D,F)
the 6 chord = minor (1,b3,5) = C minor (C,Eb,G)
the 7 chord = dim. (1,b3,b5) = D diminished (D,F,Ab)

in this key the C is the 6 chord (making it minor) and Eb is the 1 chord (making major)
if you want to expand on that bassline with other chords, you can use these other 5 chords.

if you think of this line in Bb: (Bb,C,D,Eb,F,G,A)

(scale note) (chord)(interval) (Chord) (notes)
the 1 chord = major (1,3,5) = Bb major (Bb,D,F)
the 2 chord = minor (1,b3,5) = C minor (C,Eb,G)
the 3 chord = minor (1,b3,5) = D minor (D,F,A)
the 4 chord = major (1,3,5) = Eb major (Eb,G,Bb)
the 5 chord = major (1,3,5) = F major (F,A,C)
the 6 chord = minor (1,b3,5) = G minor (G,Bb,D)
the 7 chord = dim. (1,b3,b5) = A diminished (A,C,Eb)

If you compare the two keys, you'll notice that C is still minor and Eb is still Major.
However, now the C is the 2 chord and Eb is the 4 chord.
This gives them a different function because now that kay center has changed, they play a different role.

what's really cool is the fact that you can use all of these chords in the song.
if you find ways too merge the two keys, you can get some really cool changes happening.
here's an example of what i'm talking about:

C minor > Bb major > A diminished > Ab major > G minor > F major > Eb major > D diminished
then repeat.

when you play this progression, you'll notice htat even though it's not in one key,
it still has movement to it.
there are so many ways to vary this
explore!