Step 1: Download Forge for Minecraft 1.7.10

Download the required Forge version from the download site :

Minecraft Forge Downloads

Step 2: Activate the Downloaded Jar

Click on the downloaded jar file to activate the installation process.

Step 3: Select the Client Installation

Select client install and click ok.

Step 4: Create a Forge 1.7.10 Profile

Open your Minecraft launcher and click "New Profile". Name it something memorable and then select Forge 1.7.10 from the version drop down. Click save.

Step 5: Play Minecraft with Forge

Make sure you select your new profile and click Play.