How to do a SQL Server DB schema update with zero downtime

What do you think is the best way of updating an existing SQL Server (we are using SQL Server 2014, but could update to 2016) database schema (incl its data) with zero downtime of the overall system, i.e. applications using the database?

The business requirements is to have a zero downtime of all the applications and services using the database. We could say we only do backwards and forward compatible database schema changes, e.g. adding columns, but not removing existing columns.

Of course the business would like to have a backup before we do the database change, in case something goes really wrong and we need to rollback.

The system is transactions heavy, meaning potentially thousands of transactions per second.

The applications are .net applications, where most of them run in an IIS execution container at the moment (maybe we switch to some other form like self-hosted service etc.) and exopsing their functionality through Web Services.

What would be your approach?