One of the most popular ways to customize your Minecraft game is by changing the skin (appearance) of your character. If you want to update your skin without jailbreaking your device or using the Block Launcher app, you will need Minecraft PE version 0.11.0 or higher.
Steps
For iOS Devices

Download Minecraft PE version 0.11.x+

Download the app to change Minecraft PE skins: https://itunes.apple.com/us/app/skinswitch-for-minecraft/id524911308?mt=8

Open the app, search for the skin you like, then tap "Wear Skin".

That's it! The process is incredibly simple thanks to the built-in skin support feature.
For Android Devices

Ensure you have Minecraft PE version 0.11.x+ installed

Search for "Skins for Minecraft" on Google Play and select an app from the results.

Install and open the app.

Click on the skin image that you like the most.

Tap the "Download Skin" button.

Launch Minecraft, and you'll see that your character's skin has changed!
Advice
- The new update makes it much easier to change your skin!