First of all, you'll need communication channel i.e. the way your customer can reach you. Service requests, publicly published form (e.g. on the self-service portal or service desk portal) or through Service Level management/Business Relationship Management processes - these are the common way to do it.
Once you receive customer complaint it's important to have someone to handle it. Equally important - to let the customer know that someone is working on his complaint.
An finally, internally you need to setup the procedure how to approach the complain. That would include, for example:
- whom to report that complaint has been received
- timescale (maybe hard to define general timescale to resolve complains but at least for usual complaints)
- escalation procedure
- roles and responsibilities
- depending on the services you provide - particular steps to resolve the complaint