Anything with the same Hilbert polynomial as a globally complete intersection. This is true because two fibers of the same flat projective family have the same Hilbert polynomial, and because the Hilbert scheme is connected. EDIT: Since the base of the flat family is required to be irreducible, the projective scheme must be in the same irreducible component of the Hilbert scheme as a complete intersection. I don't know any nice characterization of when this is the case.
The Hilbert polynomial of a complete intersection is easy to calculate using the Koszul complex. The coefficients will be some set of polynomials in the degrees of the hypersurfaces you completely intersect. You can set them equal to the actual coefficients of the Hilbert polynomial and check if the system of equations has a positive integer solution.
Computationally, you can just check all possible factorizations of the degree.
For curves, degree, arithmetic genus, and dimension of the embedding are the only invariants needed. The last one is important. For instance, a genus $5$ 5$, degree $8$ curve is the complete intersection of three degree $2$ hypersurfaces in $\mathbb P^4$, but not the complete intersection of any pair of hypersurfaces in $\mathbb P^3$.