public void Fill(){
DataTable dt = new DataTable();
.
.
.
gvFiles.DataSource = dt;
}

Для нестатического метода, поля или свойства требуется ссылка на объект

public Form1()
        {
            InitializeComponent();
            Run();
        }

        public static void Fill()
        {
            FileInfo[] fi = Directory.GetFiles(path).Select(x => new FileInfo(x)).OrderByDescending(x => x.LastWriteTime).Take(5).ToArray();

            DataTable dt = new DataTable();
            dt.Columns.Add("Name", typeof(string));
            dt.Columns.Add("ModifiedWhen", typeof(DateTime));

            foreach (FileInfo f in fi)
            {
                DataRow dr = dt.NewRow();
                dr["Name"] = f.FullName;
                dr["ModifiedWhen"] = f.LastWriteTimeUtc;
                dt.Rows.Add(dr);
            }
            
            gvFiles.DataSource = dt;
        }

        [PermissionSet(SecurityAction.Demand, Name = "FullTrust")]
        public static void Run()
        {
            FileSystemWatcher watcher = new FileSystemWatcher();
            watcher.Path = path;
                        
            watcher.NotifyFilter = NotifyFilters.LastAccess | NotifyFilters.LastWrite | NotifyFilters.FileName | NotifyFilters.DirectoryName;
                        
            watcher.Filter = "*.txt";
                        
            watcher.Changed += new FileSystemEventHandler(OnChanged);           
            watcher.Renamed += new RenamedEventHandler(OnRenamed);

            watcher.EnableRaisingEvents = true;
        }
        
        private static void OnChanged(object source, FileSystemEventArgs e)
        {
            Fill();
        }
}