Class function as a character
Every character is the trace of a (finite-dimensional) representation; by Maschke's theorem, every such representation is a direct sum of irreducible ones, and therefore the original character is equal to the sum of the traces of these irreducible representations, possibly with multiplicity. The multiplicities are all positive integers, because the answer "how many times does a copy of the irrep $V$ appear inside the representation $W$?" is always a non-negative integer.
Conversely, if a class function $\psi$ is equal to $\sum_{i=1}^m c_i \chi_i$ where each $c_i$ is a positive integer, then $\psi$ is the trace of the representation $\bigoplus_{i=1}^k c_i \pi_i$, where we choose each $\pi_i$ to be an irreducible representation whose trace is $\chi_i$.
That was my attempt to answer the second question. To answer the first question: it is a character when it is a character.
Less facetiously, take the inner product of your class function with each of the irreducible characters, using the normalization if the inner product which makes the irreducible characters an orthonormal basis for $\ell^2(G)$. If each of the resulting numbers is a non-negative integer, this expresses your class function as a non-negative integral combination of irreducible characters.
While one often proves that θ is a character by showing θ(1) > 0, this is not sufficient, even for differences of characters.
For instance, consider the non-abelian group of order 6 with classes 1a, 2a, 3a. It has a two-dimensional character (2,0,−1), and subtracting the trivial character (1,1,1) one gets the virtual character (integral linear combination of irreducible characters) (1,−1,−2). This cannot possibly be a character since a one-dimensional character is in fact a homomorphism, but −2 has infinite order in $\mathbb{C}^\times$. Alternatively, one already knows the one-dimensional characters: (1,1,1) and (1,−1,1).
The subset of characters within the abelian group of virtual characters is a little funny. It is not like positive and negative integers where either θ or −θ is a character. It is more like the X–Y plane, where there is the first quadrant (X>0, Y>0) and its opposite the third quadrant (X<0, Y<0), but also the second and fourth quadrants that are just weird mixtures.