1. Create a new application here: create application
2. Choose standalone applicaton and confirm app creation via SMS.
3. Fill Package namefor Android as your current package name. You can get your package name inside android manifest file, at the very begginning.
4. Get your Certificate fingerprint by executing this command in your shell/cmd:

keytool -exportcert -alias androiddebugkey -keystore path-to-debug-or-production-keystore -list -v

You can also get this fingerprint by SDK itself:

String[] fingerprints = VKUtil.getCertificateFingerprint(this, this.getPackageName());
Log.d("MainActivity", fingerprints[0]);

5. Add recieved fingerprint into your Signing certificate fingerprint for Android: field in Vk app settings (where you entered your package name)

6. Then add this to your gradle file:

compile 'com.vk:androidsdk:1.6.5'

8. Initialize the SDK on startup using the following method. The best way is to call it in the Applications onCreate method.

private static final int VK_ID = your_vk_id;
public static final String VK_API_VERSION = "5.52"; //current version
@Override
    public void onCreate() {
        super.onCreate();
        VKSdk.customInitialize(this, VK_ID, VK_API_VERSION);
}

This is the best way to initizlize VKSdk. Don't use the methid where VK_ID should be placed inside strings.xml because api will not work correctly after it.

9. Final step is to login using vksdk.

    public static final String[] VK_SCOPES = new String[]{
            VKScope.FRIENDS,
            VKScope.MESSAGES,
            VKScope.NOTIFICATIONS,
            VKScope.OFFLINE,
            VKScope.STATUS,
            VKScope.STATS,
            VKScope.PHOTOS
    };

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
    
        someButtonForLogin.setOnClickListener(new View.OnClickListener() {
            @Override
            public void onClick(View v) {
               VKSdk.login(this, VK_SCOPES); 
            }
        });
 
    }

    @Override
    protected void onActivityResult(int requestCode, int resultCode, Intent data) {
        super.onActivityResult(requestCode, resultCode, data);
        VKSdk.onActivityResult(requestCode, resultCode, data, new VKCallback<VKAccessToken>() {
            @Override
            public void onResult(VKAccessToken res) {
                res.accessToken; //getting our token here.
            }

            @Override
            public void onError(VKError error) {
                Toast.makeText(SocialNetworkChooseActivity.this,
                        "User didn't pass Authorization", Toast.LENGTH_SHORT).show();
            }
        });
    }