The easiest way to change your Twitter account picture from a phone is to use its built-in Web browser to view the Twitter website and to change your profile settings. The account picture can also be changed through the official Twitter mobile app and some third-party clients, though the exact procedure varies depending on the type of phone and software you're using. If your phone is not a smartphone and doesn't offer Web access or apps, you cannot change your Twitter account picture from it.
Launch your phone's Web browser and enter the "twitter.com" URL into the address bar. When the page appears, sign in using your Twitter account name and password (if you aren't logged in automatically).
Tap the "Me" button on the toolbar to view your own profile and recent tweets. Tap the black gear icon and then choose "Edit profile" to make changes.
Tap the "Change" link next to your current account picture. Select "Choose File" then pick an image from your phone (or take a new one using the phone camera). Tap "Save" to confirm your choice and make the change.
Tap the "Change" link under the banner photo if you wish to replace this image as well. The process of choosing or taking a photo is the same.
Any mobile browser can be used to follow the steps listed above, including the default Android browser and Apple's Safari. If the Twitter page doesn't appear, check that you have a working connection to the Internet.
The process of changing your profile picture through the official Twitter mobile app for Android and iOS is very similar -- visit the "Me" page in the app then tap the gear icon to access the "Edit profile" option. Several third-party clients, such as Tweetbot, also enable you to modify your profile picture from within the app. Check the documentation supplied with your app for more details.