Bitcoin Transaction Fees - Want to increase fees! I have my bitcoins on bittrex and I want to pay with increased fee to make sure its confirmation in the least possible time. I don't know how to do that. 

In plain words, I want to deposit Bitcoin to my local exchange but bittrex is allowing only 0.001 BTC transaction fees. I want to increase it.

I thought that I'd purchase other altcoins like NEO or LTC and transfer it to other exchange first but so far, I am unable to find any exchange that allows "increasing transaction fees".
There's no way to increase it unless the exchange themselves set the fee. However, as you can see here : https://jochen-hoenicke.de/queue/#24h

The Mempool is getting freed so the fees are getting lower. From my experience bittrex satoshi/byte for that 0.001 is around 400-500. Right now you can get away with 200 satoshi/byte so as long Bittrex actually send the coin immediately, it will confirm in 1-2 blocks.