(a) We say a character $\chi$ associated with representation $\varphi$ is irreducible if and only if the representation $\varphi$ is irreducible.
(b) In characteristic 0, the kernel of the character turns out to be exactly the kernel of the corresponding representation.
(c) This one is much more complicated. This essentially boils down to using orthogonality relations among characters. The idea is that if two characters match, then the inner product of these characters with some irreducible character will match and it turns out that this inner product is the number of times the corresponding irreducible representation appears in the representations in question. Thus because both representations decompose into the same number of copies of various irreducible representations, they must be isomorphic. To get a really good sense of what is going on here, just pick up a book on representation theory and read the chapter(s) on character theory. [For example: Martin Burrow's Representation Theory of Finite Groups (a cheap dover book) is quite accessible.]
Why is char 0 theory so different? Because much of group representation theory revolves around having an invariant inner product. The standard inner product is formed by "averaging over the group". So to compute this "average" you need to divide by the order of the group. If the group's order is not invertible, this cannot be done. So no inner product. So you can't use the standard techniques.
Do note though that characteristic p theory does go through about the same if the order of the group n is relatively prime to p. Why? Because if n and p are relatively prime, then n is invertible in a field of char p so we can form the invariant inner product etc. [All the standard techniques work.]