The first thing to think about really is the trust model that the particular application needs. The blockchain and more broadly 'distributed consensus' was created largely to solve the problem of trust and to eliminate trusting central authorities.
The trust model is important because if not properly considered you can create a solution that solves part of a problem (like letting multiple parties have a shared database or ledger), but does not solve other problems: censorship resistance, double spends, immutable ledger.
Most private blockchains will not be based on proof of work or proof of stake like all public blockchains are. Instead they will likely be based on some ring signature scheme or round robin signing. As such already some aspects of the Blockchain are lost because we immediately have a set of authorities that have been granted keys to create blocks on the blockchain. This might be fine for the particular application, but it opens the application up to certain attacks and threats.
Some specific questions to ask and understand:
- What method will be used to determine block validity (because it is probably not PoW or PoS).
- Is transaction ordering in the block important?
- What happens when block signers refuse to include certain transactions (censorship)?
- What happens when block signers collude? Can they change the immutable ledger?
- What happens when someone else gains access to a block signers key?
- Do block signers have proper financial incentives in the chain's validity and immutability?
Blockchain technology is an important new field but it was originally done as a distributed public ledger. When creating private blockchain applications there are some differences as discussed above. So keeping these differences at the forefront of thought during development will be key in making sure the resulting application is useful and results in a blockchain that has all the desired properties the parties are looking for.