Heroku is a cloud platform that supports a number of languages. .NET Core has yet to be officially supported. The options for deploying .NET Core applications to Heroku are using Third Party buildpacks or container support. It can take some digging to find a buildpack that is well maintained and working properly for the version of .NET Core you are using.
For my purposes, I am deploying a brand new ASP.NET Core Web Application using Version 1.1. In order to get this deploying successfully to Heroku, first download the Heroku CLI.
From the command prompt, login to your Heroku account to get started.
Navigate to the project’s root directory and create the app you will be deploying to.
Use the set:buildpacks command to point the app to the desired buildpack:
heroku buildpacks:set https://github.com/jincod/dotnetcore-buildpack#v1.0.4
From here all that is left is to deploy the application to Heroku.
git push heroku master
See a demo application here.