losing weight is more difficult to achieve but gaining weight is more easy, you can do it easily. Eat more foods not means that you should have all junky and fatty foods have only healthy foods, diary products, fruits and vegetables. Staying healthy is more important than gaining weight so focus on that. go to gym to gain muscles.
From your measurements, you are underweight for your height. Some people have a natural high metabolism which tends to cause them not to gain weight. Since you are underweight, you need to gain weight slowly, try to eat a Heart Healthy diet that will give you the nutrients, vitamins, and minerals your body needs, and help with possibly preventing future disease states. Suggest eating about 2000-2400 calories per day broken down to three meals a day, with snacks between meals. Heart Healthy diet which is a moderate fat diet and a moderate fat diet means = using polyunsaturated (omega 3, corn oil, nuts, etc) fats and monounsaturated (olive oil, canola oil, peanut oil, etc) fats, no saturated fats (high fat snack foods, butter, fat spreads) and no trans fats. Eat low fat meats (chicken, fish, loin of pork) or non-meat sources (soy products, nuts), low fat vegetarian combinations (rice & beans, pasta & beans, low fat cheese & pasta), low fat and substitute dairy products (1% milk, low fat/fat free yogurts, low fat soy milk/soy yogurt), use plant oils, non trans-fat margarines, and eat a lot of fruits and vegetables for vitamins and minerals. Eat daily high fiber breads and cereals (oatmeal). Suggest eating 3 meals a day constituency with drinking high nutrient drinks - Ensure, Boost or make milkshakes (milk, fruit, and ice cream blended) or yogurt smoothies (yogurt, fruit, and ice cream blended). Drink one a day for a month, then 2 a day the next month, and for the third month go up to 3 times a day to help you gain weight slowly. You have to constantly eat and drink the shakes daily to gain weight if you have a high metabolism. Hoped that helped you.
Eat more, drink more milk, and also discuss with your doctor about taking multivitamin tonic.