I think it totally depends on the type of dining out you plan to do and how long you are going to be there.
If you are looking at higher end places to eat then you need to be looking at a minimum per day of $50-75 for lunch.
For me, I usually pick up bagels, cheddar slices, ham, muffins, coffee, milk and fruit and that is what I consider an easy breakfast. That would last my family of 4 maybe two days and probably costs me in all honesty about $25-30 dollars depending on what fruit I choose. So if you worked that out over a week it would be about $85 and that is kind of one the cheap side.
Meat can also be very expensive in Hawaii and grilling is a great option on site at Aulani but they do have restricted hours for grill use. I think you cannot use them until 4ish and then I am not sure what time they are in use until, someone else here might know. I usually bring my own BBQ utensils from home, that way I dont have to wait to borrow ones from Aulani. To give you an idea of meat prices, I just pulled up a Hawaii Safeway flyer and I see this week they have Rib Eye steak on sale at $7.99/lb, Chicken legs, drumsticks and thighs for $1.49/lb as their front page specials (see the link below)
https://coupons.safeway.com/weeklyad/?store=2747&s=catalog/18_28_NOC_WC_S/2
Based on the idea that you are eating lunch and maybe some dinners outside of the villa, this loose math above and my preferences and keeping in mind you are two people, not four like us I would probably say you realistically should be planning about $600-800 for a full week. But that is only if you are looking at higher end eating or eating at the Aulani resort restaurants.
There are lots of places in the area and all over the island where you can get take out like pizza or subs, plate lunches, hot counter meals, stuff for picnics etc that are not going to cost even close to the above so it really depends on your lifestyle and how you eat.
If you have some "must do" types of places to eat then check out the menus on line and get an idea of how much it will cost from there.
One thing I always do is bring things like spices, instant oatmeal, granola bars, baggies, foil and plastic wrap etc from home so it doesnt end up pumping up my in room food bill.