How successful would someone be who completed his PhD in something such as computational biology or neuroscience but wanted to pursue post-doctoral training and beyond in computer science?
I say computer science because, presumably, dissertation work in computational X involves programming and a working knowledge of data structures and algorithms.
I believe that career changes and "reinventions" are a regular part of modern careers in highly specialized fields. Very few people will be able to work in a single domain for their entire careers, and the ability to move laterally between "adjacent" fields will be a critical skill enabling one to have greater chances and opportunities for success.
Now, that said, your chances of success in any particular job hunt will depend to some extent on how "enlightened" your future boss (or hiring staff) are: some will actively seek out anyone with the appropriate skill set and enthusiasm for the work, while others will be more focused on people who have the "direct" skills they need, assuming such people will require less training to be able to carry out their work. Neither of these positions is "better"; they just happen to coexist.