Avian influenza A(H5N1) poses a risk to public health due to its pandemic potential should the virus mutate to become human-to-human transmissible. To date, reported influenza A(H5N1) human cases have typically occurred in the lower respiratory tract with a high case fatality rate. There is prior evidence of some influenza A(H5N1) strains being just five amino acid mutations away from achieving droplet transmissibility, possibly allowing them to be spread between humans. Three of these amino acid mutations must occur within a single human host, though the exact probability of such mutations occurring is not currently known. Here, we present a mechanistic within-host infection model for influenza A(H5N1), novel for its explicit consideration of the biological differences between the upper and lower respiratory tracts. These developments enable us to estimate a distribution of viral lifespans and effective replication rates in human H5N1 influenza cases. We combine our within-host model with a viral mutation model to determine the probability of an infected individual generating a droplet transmissible strain of influenza A(H5N1) through mutation. For three required mutations, we found a peak probability of approximately 10^-3 that a human case of H5N1 influenza produces at least one virion during the infectious period. Our findings provide insights into the risk of differing infectious pathways of influenza A(H5N1) (namely the avian-human vs the avian-mammal-human routes), demonstrating the three-mutation pathway being a cause of concern in human cases. Additionally, our framework - combining a within-host infection model with a branching process model for viral mutation - is generalisable to other pathogens, allowing mutation probabilities to be more easily ascertained. Our findings are a starting point for further modelling of influenza A(H5N1) and other pathogens where differing tissue susceptibilities and human-to-human transmission is of concern.