Basic steps (taken from support forum)

1. Install the Azure SDK
2. Create a API App in Visual Studio (New Project → Web → API App (Preview))
3. Implement a controller that contains your method
4. Enable Swagger: Goto App_Start/SwaggerConfig and Remove the comment before EnableSwaggerUI (About line 176).
5. Publish to the cloud (either create an API App on portal.azure.com and select it in vs or create a new one on publish).
6. You can then select your API App from a Logic App.

• Azure API Apps for Web, Mobile and Logic Apps (2015).
• Creating connectors for your Logic Apps/Flow (Part 1–regular Connectors).
• Custom connectors, taking Azure Logic and API apps for a spin - Part 2.
• Example connectors source on GitHub.

• Integrating your Systems with Logic Apps video from Build 2016.
• Build holistic integration solutions using Azure Logic Apps and API Management video from Connect(); // 2016.
• Integrate all of the things! video from AzureCon 2015.
• Azure Logic Apps Blog