Pop in C (with barre) — alternating bass fingerstyle
In this grid, G and F are the only two chords offering a genuine alternating-bass option (both the low E and A strings are playable); C and Am each have only one bass string. The fact that F, despite being a barre chord, shares this privilege with G is worth noting: a barre chord isn't necessarily poorer in bass options than an open one.
Tips
On G and F, consciously work the alternation between the two bass strings.
On C and Am, don't look for an alternation that doesn't exist: the thumb stays on its single available string.
If F is causing trouble, it's not the alternating bass at fault but the barre itself: isolate it separately.
Chord progression — C
Bar 1
C
I
Bar 2
G
V
Bar 3
Am
vi
Bar 4
F
IV