Eat eat eat. Eat clean and eat lots. Increase the calories you intake until you start to gain weight. If you're gaining weight too fast then back it off or vice versa.
If you're just going for size and not too concerned about strength then keep the reps high ... probably 6-12reps per set. Also keep the rest fairly low ... maybe 2 or 3 minutes MAX on your big lifts and down around 1min for your accessories.
Base your workout around the big compound lifts ... bench, deads, squats, military press. I know you said upper body but for natural lifters you need to hit those big lower body exercise to place enough stress on your body to really make it grow. Supplement the big lifts with accessory work.
Sleep. Get 8hr sleep per night minimum. Your body doesn't grow in the gym. It grows when you're resting.