I realize that this question is very open-ended since it's not entirely clear what a "valid" character table is.
I would like to know whether creating a character table that has all of the required properties (such as row/column orthogonality, contains a trivial character, etc.) implies that there exists a group with corresponding modules and conjugacy classes.
(Edit) Additional context: I think my question could be simplified into as follows:
Does there exist a set of properties of a character table that allows us to find all finite groups by constructing character tables with those properties? Is this set different to the set of properties that a character table must meet when constructing it from some group?
In his famous report for the ICM, Richard Brauer proposed a number of questions on representations of finite groups. His Problem 6 asks to give necessary and sufficient criteria on a complex matrix to be a character table of a group. I think there is no satisfactory answer as of today. To give you a non-trivial challenge: Is the following matrix a character table? $$\begin{pmatrix} 1&1&1&1&1&1\\ 1&1&1&1&-1&-1\\ 1&1&1&-1&1&-1\\ 1&1&1&-1&-1&1\\ 2&2&-2&0&0&0\\ 4&-2&0&0&0&0 \end{pmatrix}$$
(the answer is in my recent paper on “character table sudokus”).