The main public Ethereum chain currently uses Proof of Work (mining). The details of Ethereum's Proof of Stake implementation have not yet been finalized, so it's not yet possible to describe exactly what you will need to "mine" it, but it will probably require a machine with good security, good uptime and a reliable network connection.
However, this probably isn't relevant to you: If you are running a private chain, you are not limited to the system being used by the main public Ethereum chain. Proof of Work is usually unsuitable for private chains, because the cumulative hashpower of the participants is likely to be low, so it would be easy for a single participant to turn on a lot more hashpower and take over the chain. Proof of Stake may well be unsuitable as well.
For private networks with known participants it is often better to use a Proof of Authority system. There is an implementation in Parity, and one is currently under development for Geth. There are also other systems like ErisDB that use the Ethereum Virtual Machine, so code you write for the public Ethereum chain will also work with those.