Every C in P3 of degree 5, genus 2 is contained in a quadric surface, Q. If Q is a cone then C is projectively normal (and passes through the vertex of the cone). If Q is smooth, C is of bidegree (2, 3) and, again, is projectively normal. So every curve with d = 5, g = 2 in P3 is projectively normal.
We will investigate the quadric Q containing C and show that Q can be a cone or a smooth quadric.
Recall that every line bundle of degree 5 on C is very ample (so no problems with existence).
If K is a canonical divisor on C, then deg(K) = 2 and K spans a line in P3. Now H - 2K has degree one and two cases are possible:
(a) H - 2K is effective
(b) H - 2K is not effective
(a) In this case |H - 2K| = {P} and two lines spanned by canonical divisors intersect in a point: the point (image of) P. So the lines spanned by canonical divisors are trisecant to C (they intersect C in a canonical divisor + P), hence they are contained in Q; since they intersect, Q is a cone with vertex (the image of) P. In conclusion, a line bundle of the form 2K+P yields a curve lying on a quadric cone.
(a') Conversely, if C lies on a cone, C passes through the vertex and any line of the rulling is a 3-secant to C. The planes through a line of the rulling cut on C a linear system of (projective) dimension 1, degree 2 which is necessarly the canonical system. We conclude that H is of the form 2K + P.
(b) This time |H - 2K| is empty and the lines spanned by the canonical divisors do not intersect. Set D = H - K, then deg(D) = 3, h0(OC(D)) = 2 and |H - D| = |K| has dimension 1. So every divisor of D spans a line (it is contained in 2 planes) and this line is contained in Q (it is a 3-secant to C). Since |H - 2D| is empty, two such lines do not meet, hence Q is smooth. Since |H - D| = |K|, we see that the divisors of |D| are cut out on C by one rulling of Q, while the other rulling cuts out on C canonical divisors.
(a') Conversely if C lies on a smooth quadric, it is of type (2,3) and the rulling cutting divisors of degree 2 yields a linear system of dimension 1, degree 2 which is necessarly the canonical system. Clearly |H - 2K| is empty in this case.
The final picture is as follows: every line bundle of degree 5 on C is very ample. Line bundles of the form 2K+P embedd C as a curve lying on a quadric cone, the other embedd C as a curve lying on a smooth quadric (this is the generic case).
The minimal free resolution is given by the minimal free resolution of a curve on a quadric.