I've got a few notes;
First of all I do belive in a good balance of macronutrients, though some will advise low carb, no carb, etc. It is all personal preference. I am for a ratio of about 40% protein, 30% carbs and 30% fat... but again, it's personal. That's just what I have found works for me.
In terms of your calories, if those are accurate (I assume you have a food scale or measuring cups and are tracking these counts accurately??) you are definitely undereating. I won't say that you have to eat X number of calories per day, but generally speaking you just aren't going to be consuming enough to meet your minimum requirements to function properly... not just in terms of overall calories but grams of protein, fats, vitamins and minerals, etc.
Most notably, I don't see any fat in that plan! Your body
needs fats to properly function on a hormonal level as well as metabolic.
I would suggest;
- Chicken Breast & Sweet Potato (I know it's not really breakfasty but I want the carbs to have energy to do my workouts.) (320 cal)
- Protein Shake (120 cal) - Is there more than just protein in that shake? After your workout your body is primed to take in some carbs in order to help quickly shuttle that protein shake to your muscles. Try pairing it with some fast digesting carbs such as fruits, rice cakes, cereal, etc. Yes, it seems like a bit diet no-no to have "simple carbs" but honestly I have come leaps and bounds since I started eating lucky charms, rice cakes and smarties with my shakes. 
- Chicken Breast & Brown Rice w/ brocoli (320 cal) - add some avocado, olive oil, udos oil, etc. to this meal.
- 3 Egg white omellete with mushrooms & onions (80 cal) Add one whole egg to the mix for some good fat and added protein. Top with some cheese, avocado, cook it in oil, etc.
- Apple (55 cal) Add some peanut butter and a piece of sting cheese, some deli meat, tuna, etc. More food in general. (Oh, and a medium-large apple is roughly 80-120 calories depending on size so you might be a bit off here)
- Oats with Soya Light milk (180 cal) - Again... fats and more protein would be good here. Add some egg whites or a whole egg, some nut butters, a scoop of protein powder, cottage cheese, etc. My last meal of the day (and again, this is just what I prefer - do what you want) is generally chicken or fish with asparagus or some other fiberous veggies with fish oils, cheese, olive oil, olives, nuts, or some other fat. If I am in a mood for something sweet, I generally have cottage cheese with peanut butter or something like that.
Don't be afraid of fats! Start bumping up your calories - get 1200 a day at least for the next week and then keep evaluating and raising them slowly; 100 or so calories per day for a week at a time - this should help you avoid the all-to-common and sometimes scary water retention that comes from an increase in calories. I PROMISE you, it is water weight and extra food weight... it would be damn near impossible for you to gain fat on 1200-1500 calories a day.
Also, just to hammer the point home; why are you strength training if you aren't eating enough to fuel those muscles? You need food to build muscle, which is the whole point of lifting weights. You are spinning your wheels and wasting your time if you aren't giving your body what it needs.
Lastly, yes, diet is INFINITELY more important that exercise in terms of weight loss or gain. Whatever your goal is, you need to eat in accordance with that goal to get where you want to go.
