Often I use @Ajax.ActionLink then setup AjaxOptions or use regular JQuery Ajax function to check for success. This question is specific to the following format:
using (Html.BeginForm("DoSomeAction", "ControllerName"))
Html.BeginForm renders a normal form tag without any ajaxified form submit behavior. So when you click the submit button, It submits the form to the server. Browser will handle the response from the server ( show the content/ redirect to a new GET action etc)
How to check appropriately for the success of this call??
This depends on what you want to do. If you want to execute some sever code, you can do that in the HttpPost action method to which the form data is submitted.