I am trying to write c# code to add custom webpart programmatically to SharePoint 2013 page.
I tried searching net and stackoverflow but couldn't find any proper article or blog which can help me to achieve this.
You could use the the Webpart XML content to add a Webpart in page using CSOM, below is my sample for you. It's adding a document library to the page:
var wpXML = @"<?xml version='1.0' encoding='utf-8' ?>
<webParts>
<webPart xmlns='http://schemas.microsoft.com/WebPart/v3'>
<metaData>
<type name='Microsoft.SharePoint.WebPartPages.XsltListViewWebPart, Microsoft.SharePoint, Version=16.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c' />
<importErrorMessage>Cannot import this Web Part.</importErrorMessage>
</metaData>
<data>
<properties>
<property name='ListUrl' type='string'>Shared%20Documents</property>
<property name='MissingAssembly' type='string'>Cannot import this Web Part.</property>
</properties>
</data>
</webPart>
</webParts>";
var page = ctx.Web.GetFileByServerRelativeUrl("/sites/test/SitePages/wiki1.aspx");
var wpm = page.GetLimitedWebPartManager(PersonalizationScope.Shared);
ctx.Load(wpm);
ctx.ExecuteQuery();
var importedWebPart = wpm.ImportWebPart(wpXML);
var webPart = wpm.AddWebPart(importedWebPart.WebPart, "mainContent", 0);
ctx.ExecuteQuery();