Lots of different opinions here. I'll try and give you a more medical side of things.
I really need to stress that you should NOT gorge yourself on fatty foods. For your own safety and the sake of your body.
It's not just about calories, despite what people will tell you. You can eat 4000 calories in fatty cheeseburgers each day and still not gain weight, it's about a consistent and required mix of essential bodily needs, not just 'upping your calorie content'. The body burns fat incredibly quickly, so if all you're taking in are high fat foods (fast food, processed foods) then all your body will have to burn is pure fat. Even if you're doing little to no exercise that will burn very, very quickly, resulting in little to no weight gain.
Imagine your body like a bonfire and you want the fire to grow. Fatty foods would be like petrol, it would cause a big flame but die out within a few seconds, it's not consistent, and not reliable for a decent weight gain. Further to this high fat foods will cause you a whole host of health issues down the line. You may be young, but the damage it can do to your heart, liver, kidneys is not to be ignored. I cannot stress this enough, you are not immune to these issues due to your age.
Your diet should be no different to someone who wants to lose weight. A mix of proteins, carbs, etc, the only difference is you should be eating more of them. You should be looking to gain only a couple of pounds a week, anything more than that and you'll be gaining fat in dangerous areas (heart, liver and other systemic areas which can cause a host of issues).
Do not stop exercising. While avoiding extensive cardio will help the weight gain, this can be substituted with simple light weight training, etc.
But the most important point I want to stress is if you're serious about this, and I mean really serious, then go and see a doctor. If you pick and choose from the advice in this thread you run the risk of harming your body, though most likely there will be no progress.
