Does there exist a non-hyperelliptic Riemann surface with automorphism group $C_2\times A_4$? Does there exist a non-hyperelliptic Riemann surface of genus 5 with automorphism group $C_2\times A_4$?
 A: One thing that should be made more precise is whether you are asking that the automorphism group equals $C_2 \times A_4$ or if it is enough that it contains $C_2 \times A_4$. I'm assuming you mean the former. Then it seems that the answer is no. In the paper "The locus of curves with prescribed automorphism group" (K. Magaard, T. Shaska, S. Shpectorov, H. Voelklein, http://arxiv.org/abs/math/0205314) one finds a classification of loci in the moduli space $M_g$ of curves with "large" automorphism group for small genus, meaning that the order of the automorphism group is at least $4(g-1)$. In this classification we see that there is a unique 1-dimensional family in $M_5$ of curves with full automorphism group $C_2 \times A_4$. By combining with the results of "Some special families of hyperelliptic curves" (T. Shaska, http://arxiv.org/abs/1209.1867) it seems that all those are hyperelliptic.
PS - I remember that there is an error in one of the tables in the paper of Kuribayashi and Kimura referenced by Igor Rivin - proceed with caution. 
A: There is a non-hyperelliptic (fixed-point free) involution of the surface of genus $5,$ with the quotient a surface of genus $3.$ Further, $A_4$ does come up as the automorphism group thereof, see
S. Allen Broughton, MR 1090743 Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1991), no. 3, 233--270.
So, modulo some care, the answer seems to be YES.
ADDITION As pointed out by Noam in the comments, the above is not quite satisfying. The simplest reference is one to the paper of Kuribayashi and Kimura
Akikazu Kuribayashi and Hideyuki Kimura, MR 1068416 Automorphism groups of compact Riemann surfaces of genus five, J. Algebra 134 (1990), no. 1, 80--103.
Unfortunately, that paper's notation is somewhat hard to penetrate. On the other hand, the way they describe the automorphism group (conjugacy class in $GL()$ should make it easy to check that the relevant surface is not hyperelliptic.
