In cells, the breakdown of individual amino acids to simple metabolites (glucose precursors, fatty acid precursors, and ammonia) are complex multi-step pathways, so there is no simple mechanism to describe the breakdown of all amino acids. It differs for each amino acid (or at least group of similar amino acids). Any good biochemistry text should have detailed reation sequences showing the degradation of each amino acid.
It's also worth noting that amino acids are rarely used to form fatty acids, since most amino acid breakdown occurs during times of high metabolic need. Usually the resulting metabolites from amino acid breakdown will be converted to glucose (from the glucogenic amino acids) or ketone bodies (from the ketogenic amino acids).