Chrome has some built-in Flash support, which is great, but sometimes it just doesn't work—and it doesn't work for everything. Unfortunately, the Flash Player plugin was discontinued by Adobe back in the days of Ice Cream Sandwich (Android 4.0), and you can't get it on Google Play.
However, you can still install the latest version of Adobe Flash Player on your HTC One, albeit without official support. This will let you do things like watch Amazon Prime instant videos and even play a few Flash games.
Since this isn't available on Google Play, you'll need to grant your phone the ability to install apps from third-party sources. To do so, go to Settings -> Security and make sure Unknown Sources is checked.
If you downloaded it on your phone, you can just tap the notification in your Notification tray when it finishes downloading, or find it in your Download folder.
Depending on your current ROM, the normal browser might work with the Flash plugin. For those that don't, you can download an alternative, and I use Boat Browser (free on Google Play).
In Boat Browser, go to Settings, Settings again, Page Content Settings, and Enable flash/plugins. Choose Always on, and Flash content will load automatically when you visit pages. On Demand means you have to tap content before it loads.
Flash is easy to install if you want to play some games without having to download apps such as Kongregate Arcade, or so you can finally watch Amazon Instant Videos, since there is no dedicated app for it yet.
If you don't like Boat Browser, you can also try out Firefox for Android (also free on Google Play).