Yes, it worked pretty much in the exact dual way: $C \oplus k$ has a co-algebra structure given by:

$$ \Delta(c + x)= \Delta(c) + c \otimes 1 + 1 \otimes c + x ( 1 \otimes 1) $$

where $c \in C$, $x \in k$ and $1$ denotes the units of $k \subset C \oplus k$

This coalgebra structure has a counit given by the projection $C \oplus k \rightarrow k$ and is universal amongst counital coalgebra with a morphism to $C$.