Install the StackExchange.Exceptional.AspNetCore nuget package via:

Install-Package StackExchange.Exceptional.AspNetCore

If setting up a console application, I encourage you to check out the .NET Core Console sample project, it has all of the below in a proper context.

You can either configure things via a config file, for example appsettings.json:

{
  "Exceptional": {
    "Store": {
      "ApplicationName": "Samples (ASP.NET Core)",
      "Type": "SQL",
      "ConnectionString": "Server=.;Database=Local.Exceptions;Trusted_Connection=True;"
    }
}

...and hook up that configuration at startup (this is an example - there are many ways to do this):

var config = new ConfigurationBuilder().AddJsonFile("appsettings.json").Build();
var exceptionalSettings = config.GetSection("Exceptional").Get<ExceptionalSettings>();
Exceptional.Configure(exceptionalSettings);

Or, you can opt to configure entirely through code instead:

 Exceptional.Configure(new ExceptionalSettings() { DefaultStore = new SQLErrorStore(_connectionString,"My Application") });

...then to log exceptions:

exception.LogNoContext();

If you want to store some custom key/value style data with an exception, you can use .AddLogData extension method, for example:

exception.AddLogData("Example string", DateTime.UtcNow.ToString())
         .AddLogData("User Id", "You could fetch a user/account Id here, etc.")
         .AddLogData("Links get linkified", "https://www.google.com");

...and these pairs will appear on the error detail screen in a "Custom" section and in the CustomData dictionary of Exceptional.Error.