Summary:
Installing an email signature in Salesforce involves creating a Letterhead and then assigning that new letterhead to an Email Template which has your HTML signature inside it.
Step 1:
Log in to Salesforce and head over to the Settings section by clicking the cog in the top right corner and then clicking Setup.

Step 2:
From the menu on the left, under Administration, click Email > Classic Letterheads.

Step 3:
Click on New Letterhead.
Ensure you tick the Available For Use box.
Fill in the Letterhead Label and Letterhead Unique Name.
Click Save.


Step 4:
You will need to edit the Header Properties, Top Line, Middle Line, Footer Properties and Bottom Line and ensure they all have a Background Color or Line Color of #FFFFFF. Also, check to ensure that all of them have a Height (pixels) of 0.
Make sure you do not edit the Body Colors (middle section).
When you’re done, click Save.

Step 5:
From the menu on the left, under Administration, click Email > Classic Email Templates.

Step 6:
From the folder dropdown, select My Personal Email Templates.
Click on New Template.

Step 7:
When choosing the type of email template to create, select “HTML (using Classic Letterhead)” and click Next.

Step 8:
Ensure the folder is My Personal Email Templates and that Available For Use is ticked.
Enter an Email Template Name and Template Unique Name.
Select your Classic Letterhead from the dropdown list.
Email Layout should be set to Formal Letter.
Encoding should be set to Unicode (UTF-8).
When you’re done, click Next.

Step 9:
In this step, you’re required to type in a subject for the new email template. This is the subject that will be applied to your email when you use this template.
Paste your email signature (Ctrl + V) into the middle box.
Click Next.

Step 10:
This section allows you enter the text-based information for your email signature in case your email is converted to plain text format in transit.
Click Copy text from HTML version.
Click Save.

Step 11:
Your Salesforce email signature is now installed!