Changing your picture on Twitter is fairly simply.
Just go to your profile page (hit the "me" icon), then select "edit profile" below your image to the right (next to the envelope icon you use to access direct messages).
This will bring up edit options for the several parts of your profile, including your photo.
You will now see a pencil icon on the top right corner of your image. Click into that and a drop down menu will appear as you can see below:
