We have been asked this quite a few times only to find out that Instagram does not really support it but there is a couple of options:
1. Download an alternative app for Instagram that will let you sign into your alternative account, the down side of this is there really isn’t any that are as simple and easy to use as Instagram
2. This is our preferred method, in Android 5.0 Lollipop user accounts is now available on smart phones as well as tablets. So we simply added another user, downloaded Instagram in that user account and then signed in to our alternative. It works very easily because if you want to use your second account – just swipe down the main menu and then tap the other account – you’ll then be able to open Instagram as the other user without having to log out/log in – Happy Days!
On the topic of Instagram, in July they released Layout (Android and iPhone) you no longer need to use InstaPic or similar to get those multiple frame shots – we’ve been using the Layout App and it works really well.