Since Better Twitch TV added emojis to Twitch chat, there hasn't been a simple way to use them. You had to either go on twitter, copy them, go on your phone to post them, now you can easily do it all, without ever leaving Twitch.
Open the emoji menu and you can easily search for the emojis, or just scroll til you find the one you want.
* Button to active emoji may not appear, refresh if this happens.
* Menu will stay if you change page (inside twitch.tv), again refresh if this happens.
* If whispers are open and you use the emote menu, it will write in both whisper input and chat input, furthermore it will remove anything in the current chat input.
---- All these will be fixed in next update <3