Example
ClientContext oClientContext = new ClientContext("http://MyServer/sites/MySiteCollection");
File oFile = oClientContext.Web.GetFileByServerRelativeUrl("Default.aspx");
LimitedWebPartManager limitedWebPartManager = oFile.GetLimitedWebPartManager(PersonalizationScope.Shared);
string xmlWebPart = "<?xml version=\"1.0\" encoding=\"utf-8\"?>" +
"<WebPart xmlns:xsi=\"http://www.w3.org/2001/XMLSchema-instance\"" +
" xmlns:xsd=\"http://www.w3.org/2001/XMLSchema\"" +
" xmlns=\"http://schemas.microsoft.com/WebPart/v2\">" +
"<Title>My Web Part</Title><FrameType>Default</FrameType>" +
"<Description>Use for formatted text, tables, and images.</Description>" +
"<IsIncluded>true</IsIncluded><ZoneID></ZoneID><PartOrder>0</PartOrder>" +
"<FrameState>Normal</FrameState><Height /><Width /><AllowRemove>true</AllowRemove>" +
"<AllowZoneChange>true</AllowZoneChange><AllowMinimize>true</AllowMinimize>" +
"<AllowConnect>true</AllowConnect><AllowEdit>true</AllowEdit>" +
"<AllowHide>true</AllowHide><IsVisible>true</IsVisible><DetailLink /><HelpLink />" +
"<HelpMode>Modeless</HelpMode><Dir>Default</Dir><PartImageSmall />" +
"<MissingAssembly>Cannot import this Web Part.</MissingAssembly>" +
"<PartImageLarge>/_layouts/images/mscontl.gif</PartImageLarge><IsIncludedFilter />" +
"<Assembly>Microsoft.SharePoint, Version=13.0.0.0, Culture=neutral, " +
"PublicKeyToken=94de0004b6e3fcc5</Assembly>" +
"<TypeName>Microsoft.SharePoint.WebPartPages.ContentEditorWebPart</TypeName>" +
"<ContentLink xmlns=\"http://schemas.microsoft.com/WebPart/v2/ContentEditor\" />" +
"<Content xmlns=\"http://schemas.microsoft.com/WebPart/v2/ContentEditor\">" +
"<![CDATA[This is a first paragraph!<DIV> </DIV>And this is a second paragraph.]]></Content>" +
"<PartStorage xmlns=\"http://schemas.microsoft.com/WebPart/v2/ContentEditor\" /></WebPart>";
WebPartDefinition oWebPartDefinition = limitedWebPartManager.ImportWebPart(xmlWebPart);
limitedWebPartManager.AddWebPart(oWebPartDefinition.WebPart, "Left", 1);
oClientContext.ExecuteQuery();
