Check- in & check-out files using client object model

To check-out:

//get the connection
ClientContext ctx = new ClientContext("http://sitename");

//get the home page
File home = ctx.Web.GetFileByServerRelativeUrl("/SitePages/home.aspx");
ctx.load(home);
//check whether file is already checkout. If not then only do the checkout. 
if (home.CheckOutType == CheckOutType.None)
{
   home.CheckOut();
}

To check-in:

//Only checkin if the file is already checkout. 
if (home.CheckOutType != CheckOutType.None)
{
   home.CheckIn(string.Empty, CheckinType.MajorCheckIn);
}