Load the xml into an XElement:
var xml = XElement.Load("test.xml");
Execute XPath to select the SN elements in the cats with id computer: (+ put them in a list)
var snValues = xml.XPathSelectElements("//cat[@id='computer']/item/SN")
.Select(x => x.Value).ToList();
Required usings:
using System.Linq;
using System.Xml.Linq;
using System.Xml.XPath;