I think for this u have to open online wallet with ur paper wallet keys and then u will be able to send your litecoins from that wallet to bittrex wallet. Wait for more suggestions as I may be wrong
I'd defenatly discourage the use of an online wallet...
Many desktop wallets, like electrum for litecoin, allow you to sweep a private key, or to restore a wallet using your private key.
There are also some mobile wallets that even allow you to scan the paper wallet's private key's QR code... But i wouldn't hold my funds on a mobile wallet for prolonged periods of time either...