Matrix of a representation from character theory we have learnt in class that a representation (of a finite group $G$) is completely determined by the characters on its conjugacy classes. First of all, I know that the characters can be used in a lots of ways:
(1) By the orthogonality relation for characters, we can directly see if a representation is irreducible just by looking at the characters.
(2) If we have an arbitrary representation and the complete character table of all the irr.reps, we can directly derive the decomposition of the given rep.
But I have the following question: Is it possible to derive the basis representation of a given representation in a given basis just by looking at the characters? I mean if a representation is ''completely determined by its characters'', then it should be possible. 
 A: I do not understand your question very well: how is the representation "given"? 
I will start by saying that there is no easy method to recover a representation from its character: in fact, for a lot of groups the character table can be computed using orthogonality relations and other character properties, but it is much harder to identify the representations corresponding to some rows.
I'll assume we are in characteristic zero. What do you mean by "a given representation"? If it is given explicitly then your question does not make sense, so I don't think it means that. If its character is given, then you can use the inner product to compute its decomposition as a linear combination of irreducible characters:
$$ \psi = \sum_{\chi \in {\rm Irr}(G)} \langle \psi, \chi \rangle \chi$$
and then, if you know the matrices corresponding to each irreducible character (which you often don't) you can easily compute the matrix corresponding to $\psi$, as the character decomposition corresponds to the decomposition into a direct sum of simple representations.
