How would I say my name when speaking Japanese?
Would I use the Japanese pronunciation or the pronunciation of my home country? For example would I say "Watashi wa Gaburieru desu" or "Watashi wa Gabriel desu".
Your choice. For some documents the katakana is required. For others, it is not. How you choose to represent yourself is an individual choice. You can attempt to inspire people to use the actual pronunciation of your name or you can simplify it for yourself and others. Ball is in your court.
Like BJCUAl said, it's up to you. I went to Japan and many of them preferred using my nickname as my former Sensei gave me kanji for my name so most of them were overjoyed at writing my name in kanji. But katakana or no it's all up to you. I was lucky in that my name works perfectly in Japanese.
If you wanna be extra polite about it, you could say Gabriel to moushimasu. と申します。
When I was in Japan I found that a lot of people had trouble understanding names that contained R and L noises, which could be a little awkward for people. Sometimes I'd give both pronunciations, but I leaned towards the Japanese pronunciation so that people were more likely to be able to understand the sounds I was making.