It is possible. However, there are a few difficulties. Normally, cryptocurrencies are based on a proof-of-work (PoW) scheme which involves computing data which is hard to find but easy to verify. A collision-resistant hash function such as SHA-256 or Scrypt is an example of this, and is used as PoW for a majority of existing cryptocurrencies. Unfortunately, there isn't really a cryptographic way to verify that someone has been legitimately tutoring for a period of time, so the currency will have to be issued by a central authority (i.e. you).
This means that you will have to hold a significant portion (perhaps all) of the coins in the beginning (in cryptocurrency circles, this is called a "premine"). If you hold all the coins, then you would be well-advised to choose proof-of-stake (PoS), where people leave their wallets open and are rewarded coins at a rate dependent on their current holdings (called "minting"), which incentivizes people to secure the network. (Otherwise, there would be no miners to process transactions since there's no incentive to do so, and the coin would be dead.) Alternatively, you can premine something like 10% or 50% of the coins, leaving the rest to be mined just like Bitcoin or Litecoin on a decreasing schedule. Here PoW miners are willing to secure the network for you because of the block rewards.
The issue with premined cryptocurrencies is that many people don't trust a single entity with all those initial coins, so they are unlikely to gain much value. Thus your tutors might not be willing to work for essentially worthless coins. And besides, since you're going to have a lot of central control due to your premine, you might as well just create an in-house currency, kept track of on your servers, rather than a cryptocurrency. Maybe eventually have a online store where users can buy stuff using that currency.
Another alternative is to ask for Bitcoin donations (possibly from the students), and then pay tutors with that (or even have students tip tutors directly).