Sam,
You may wish to change the UNC path. When I ran your script on my test server, adding the complete UNC path works:
declare @script nvarchar(4000)
set @script = N'
exec master.dbo.xp_backup_database
@database = N'LiteSpeedLocal',
@backupname = N'LiteSpeedLocal - Full Database Backup',
@desc = N'Full Backup of LiteSpeedLocal on 5/28/2013 7:59:45 AM',
@compressionlevel = 2,
@filename = N'\\Server_name\c$\Backup\2005\LiteSpeedLocal_201305280759.bak',
@init = 0,
@with = N'STATS = 10''''
exec (@script)
The SQL Server service account must have read/write/create permissions to the destination folder.