Duplicate Check offers a REST API which can be used to search or insert/update records in Salesforce via the Data API.
The REST Data API lets you create records without creating duplicates. When a record is added via the Data API this record is directly checked for duplicates. If duplicates are present, it will update the existing record instead of creating a duplicate record.
The API documentation can be found via swaggerhub.
You need to authenticate yourself before you can use the REST API. The authentication works via the standard Salesforce Rest Authentication methods. Read more about this in the REST API Developer Guide.
The user you use to access the DC REST API needs to have the DC Permission Set in order to use the API.