How to get profile picture in android using Twitter kit,

From the official doc:

You can obtain a user’s most recent profile image from GET users/show. Within the user object, you’ll find the profile_image_url and profile_image_url_https fields. These fields will contain the resized “normal” variant of the user’s uploaded image. This “normal” variant is typically 48x48px.

By modifying the URL, you can retrieve other variant sizings such as “bigger”, “mini”, and “original”.

Following the code:

TwitterApiClient twitterApiClient = TwitterCore.getInstance().getApiClient();
twitterApiClient.getAccountService().verifyCredentials(false, false, new Callback<User>() {
    @Override
    public void success(Result<User> userResult) {
        String name = userResult.data.name;
        String email = userResult.data.email;

        // _normal (48x48px) | _bigger (73x73px) | _mini (24x24px)  
        String photoUrlNormalSize   = userResult.data.profileImageUrl;
        String photoUrlBiggerSize   = userResult.data.profileImageUrl.replace("_normal", "_bigger");
        String photoUrlMiniSize     = userResult.data.profileImageUrl.replace("_normal", "_mini");
        String photoUrlOriginalSize = userResult.data.profileImageUrl.replace("_normal", "");
    }

    @Override
    public void failure(TwitterException exc) {
        Log.d("TwitterKit", "Verify Credentials Failure", exc);
    }
});

For further information refer to Twitter API Documentation | Profile Images and Banners

As of Nov 2016. This works. There is a change in the implementation of verify credentials.

Call<User> user = TwitterCore.getInstance().getApiClient().getAccountService().verifyCredentials(false, false);
           user.enqueue(new Callback<User>() {
                       @Override
                        public void success(Result<User> userResult) {
                            String name = userResult.data.name;
                            String email = userResult.data.email;

                            // _normal (48x48px) | _bigger (73x73px) | _mini (24x24px)
                            String photoUrlNormalSize   = userResult.data.profileImageUrl;
                            String photoUrlBiggerSize   = userResult.data.profileImageUrl.replace("_normal", "_bigger");
                            String photoUrlMiniSize     = userResult.data.profileImageUrl.replace("_normal", "_mini");
                            String photoUrlOriginalSize = userResult.data.profileImageUrl.replace("_normal", "");
                        }

                        @Override
                        public void failure(TwitterException exc) {
                            Log.d("TwitterKit", "Verify Credentials Failure", exc);
                        }
                    });

Found the answer.

There is a callback which returns you the User object.

TwitterCore.getInstance().getApiClient().getAccountService().verifyCredentials(true, true, false).enqueue(new Callback<User>() {
    @Override
    public void success(Result<User> result) {

    }

    @Override
    public void failure(TwitterException exception) {

    }
});

On the success branch, you can get the User object by calling:

User user = userResult.data;

And from this object you can get all the information form the user. For the profile image:

String profileImage = user.profileImageUrl;