According to the Microsoft documentation (found here: https://learn.microsoft.com/en-us/azure/application-insights/app-insights-api-custom-events-metrics), the MetricManager class can be used in the following way:
// Initially:
var manager = new Microsoft.ApplicationInsights.Extensibility.MetricManager(telemetryClient);
// For each metric that you want to use:
var metric1 = mgr.CreateMetric("m1", dimensions);
// Each time you want to record a measurement:
metric1.Track(value);
I updated the Application Insights for ASP.NET library to version 2.4 (which is still in pre-release, by the way). However, the MetricManager class is not exposed publicly. When I decompose the dll, it looks like the MetricManager class is internal. Is this because the library hasn't officially been released yet?