As a fan of Asian TV Shows and Movies, I like to share the way I watch them on Amazon Fire TV Stick. The Apps are free but you might need a VPN if you want to access to the contents that are not available in your region.
The first choice would be Viki.
It has a good collection of Asian TV Shows from China, Korea, Thailand and other Asian countries as well. The best part of it is you always have the English Subtitle available.
They offer both FREE and Subscription. Most of the TV Shows are free to watch, except some of them you would need to pay for the subscription.
Keep in mind that there are some advertisement if you stay with the FREE subscription. But usually they are not too long and often you will get between 1 to 3 advertising per episode. I hope you can tolerate it, if not, you can purchase Viki Pass at $4.99 per month.
The second choice would be IcDrama Kodi Addon.
You need to install Kodi first on Amazon Fire TV Stick. If you are new, here is a good tutorial on how to install Kodi on FireStick.
Some of the TV Shows in IcDrama has English subtitle but for some TV Shows, you need to find and add the subtitle manually in Kodi.
The good source to get the subtitle for Chinese TV Shows would be Subscene. You can extract the subtitle and store in on a local shared folder. Then from Kodi, you can browse the subtitle that you stored on the shared local folder. Let me know if you need help on that but I imagine it is not difficult to browse the subtitle.
Remember to set the folder for the Subtitle in Kodi setting.