Edit: oops, the scripts didn't show - fixing
You'll need to reference the Ajax scripts shown here:
<script src="../../Scripts/MicrosoftAjax.js" type="text/javascript"></script>
<script src="../../Scripts/MicrosoftMvcAjax.js" type="text/javascript"></script>
<script src="../../Scripts/AjaxLoadedContentScriptFix.js" type="text/javascript"></script>
I don't recall what that last script is, but I'm using it in my app so it probably can't hurt :)
Then you'll need to use Ajax.BeginForm in your view to post a form asynchronously. It takes an AjaxOptions parameter that will allow you to specify the update target, success callbacks, etc. Example:
<% Ajax.BeginForm("FormName" , new { id = Model.SomeProperty },
new AjaxOptions { UpdateTargetId = "MyDivToUpdate", OnSuccess = "onSuccess" }); %>