You can't really, unless all your Instagram contacts also use the same handle on Twitter.
But if the people you're tagging on Instagram also have a linked Twitter account, even if it's a different name, the tweet will convert the usernames as necessary.
If the @mentioned user has connected Instagram to Twitter:
The user’s Instagram username will appear in the photo caption
The user’s Twitter username will appear in the tweet
If the @mentioned user has not connected Instagram to Twitter:
The user’s Instagram username will appear in the photo caption
The @ sign will be removed from the username when shared to Twitter
Instagram's own help page on Creating an Account & Username says that you need to download the Instagram app on your device in order to register an account.
Mashable's The Beginner's Guide to Instagram also say that "Instagram has always been an almost exclusively mobile platform. Therefore, you must download the iPhone or Android app to your device in order to register an Instagram account."
It's worth noting that Facebook now own Instagram so it's not unlikely that in future you'll be able to login directly using your Facebook account, but as of yet there's no indication that will be the case.
Update
Instagram now allow you to create an Instagram account from a computer:
Go to instagram.com.
Enter your email address, create a username and password or click Log in with Facebook to sign up with your Facebook account.
If you register with an email, click Sign up. If you register with Facebook, you'll be prompted to log into your Facebook account if you're currently logged out.
Best Answer
You can login to Instagram website using any browser from your computer or mobile and post whatever you want post. You don't need to download app.