The character of a finite group representation seems to carry most of the information that the representation itself has, because the character identifies the representation up to isomorphism, by virtue of the orthogonality of irreducible characters, and the semisimplicity of group representations (over characteristic $0$). In the very first paragraph of the Wikipedia page on Character theory (https://en.wikipedia.org/wiki/Character_theory), it says "Fröbenius initially developed the representation theory of finite groups entirely based on the group characters, and without any explicit matrix realization of representations themselves."
My question is this: Is it possible to define group characters intrinsically, that is, without talking about representations at all? A weaker question is: is it possible to recognise some non-trivial class functions as definitely being or not being characters without first calculating the character table and using the orthogonality relations? If there are answers for Lie groups or locally compact groups, they would also be welcome.
Finally, if the answer is in the negative, what is the Wikipedia page trying to say exactly? How did Frobenius work with group characters without the matrix representations (and what does this even mean)? If someone could reference any exposition of his development, that would be very helpful.