After a mutation at a genetic locus the character of an organism changes due to the change in :-

A gene codes for a polypeptide through the genetic code: DNA $\rightarrow$ mRNA $\rightarrow$ protein.
A mutation changes the base sequence of the gene. This changes the codons in the mRNA, and therefore the amino acid sequence of the protein made from it.
A change in the amino acid sequence changes the structure (and so the function) of the protein, e.g. an enzyme or a structural protein, and this changes the character of the organism.
Example: in sickle cell anaemia, a single base change (GAG to GUG) replaces glutamic acid with valine at position 6 of the $\beta$ chain of haemoglobin. The altered haemoglobin structure makes RBCs sickle shaped.
The overall pattern of DNA replication, transcription or protein synthesis does not change.
So the character changes due to the change in protein structure.