Apple announced iOS 10 during WWDC 2016. It looks amazing so I wanna install iOS 10 on my iPhone 6s. But I don't have a developer account of Apple.
Without any research on the matter I would say: you can't.
That's exactly what developer account are for: so a new software / a new system can be released to a small group of expert users. While the software / system may already look ok in a presentation, it probably still is way too buggy for the average user. So either be patient, or become a developer.
Also, Apple has released the last few major OS upgrades (for both Macs and iPhones/iPads) as Public Beta some months before the official release. This will allow you do get your hands on these new versions some time between now and whenever-the-release-date-will-be.