How does death from a fatal dose of aspirin work? Would you die from liver or kidney failure?

Respiratory failure. Too much Aspirin interferes with the last part of your aerobic metabolism of energy, cytochrome c. It uncouples this area and one has to resort to anaerobic metabolism. Severe metabolic acidosis will ensue, the respiratory system will try to compensate but will fail and combined acidosis ensues and lack of energy just will not sustain life.