Use groupby.rank function.
Here the working example.
df = pd.DataFrame({'C1':['a', 'a', 'a', 'b', 'b'], 'C2': [1, 2, 3, 4, 5]})
df
C1 C2
a 1
a 2
a 3
b 4
b 5
df["RANK"] = df.groupby("C1")["C2"].rank(method="first", ascending=True)
df
C1 C2 RANK
a 1 1
a 2 2
a 3 3
b 4 1
b 5 2