Here are the steps you should execute

1. Get the XML for webpart
2. Load Limited Webpart Manager object
3. Add web part.

You can find details on https://msdn.microsoft.com/en-us/library/office/hh185010(v=office.14).aspx#sectionSection1

Add an existing WebPart Using the Sharepoint client object Model

Here are the steps you should execute

1. Get the XML for webpart
2. Load Limited Webpart Manager object
3. Add web part.

You can find details on https://msdn.microsoft.com/en-us/library/office/hh185010(v=office.14).aspx#sectionSection1

Add an existing WebPart Using the Sharepoint client object Model