Step One
First, have your email address forwarded to your Gmail address.
To set up forwarding you must be in your forwarding email's forwarding settings. Some email services require you to have your provider set up forwarding.
Step Two
Go to: gmail.com
You might see this first:

(If you don't see the above image, then go straight to Step Three)
If you see the above image, then click sign in (or create a Gmail if you need to)
Once you click sign in, the screen will change to image in step three.
Step Three
Put in your email or user name (username is part of gmail address before the @gmail.com)
and password. Gmail will open.
Step Four
On the right hand top side of the screen select and click the settings cog:

Step Five
The Menu Box opens, then Select and click 'settings'.
Step Six
Screen changes to setting screen, then select 'Accounts and Imports' and under that next to the right of the email whose settings you want to change select 'edit info'.
Step Seven
A pop up screen comes up, then you will see the name and email address you are changing settings for, and make sure 'Treat as an alias' is unchecked, then click 'Next Step'.
Step Eight
The window changes to this screen, then add the SMTP Server Address (provided by your email provider), user name (usually your full email) and email password. Then click 'Add Account'.
Step Nine
The window changes to the screen below. An email will be sent to the email address you are adding.
Log into the email provider of the email address you are adding and click the link it gives you to authorize your Gmail to use it in Gmail.
Then come back to this popup box and click verify (you might have to enter the code sent to you in the email of the email address you are adding in the slot next to the 'verify' button). This pop up box closes.
Step Ten
Step Eleven (optional)
If you want the email address you are added to Gmail to be the default email when you compose outgoing email, then go to 'Accounts and Imports' (Same screen as above). Then to the right of the email you added to Gmail click 'make default'.