The first thing to do would be have an official documented start, somewhere on the internet where it displays a date and time. If you decide to start logging on Tamatalk, that would cover documentation and also the more consistently you do so (not necessarily every day), the more credible your record would be. The reason consistency is more credible is because it gives more evidence to confirm or deny your claim. CthulhuSyd idea is good, but I would recommend you also take a picture of the screen that displays the generation number, since character and time can be potentially faked without messing with the program, but the generation screen cannot. Also, make sure in these pictures you show the whole Tamagotchi, as the shell design will also add consistency to your record.
The key to credibility is giving a sensible reason for why there should not be any doubt, not so much as the potential the record could have been tampered with. However, the more you ground your documentation in what cannot be changed, the easier credibility will come. Also, evidence that is more credible is evidence that is unique because what is unique cannot be faked.