My rescued puppy is weak and underweight. How to get him healthy?

Awww. It's very nice of you to take on this type of responsibility. The first thing you should do is to take him into the vet for an initial examination. They will be able to make sure there is nothing seriously wrong, versus simply malnutrition and dehydration. If this is the case, Champ will likely require something to eliminate gastrointestinal parasites. He belly looks very round in that picture, which in young dogs is usually associated with worms. This can be a very common cause of weakness and malnutrition, and the nutrients that he is consuming will not be properly absorbed until the parasites are eradicated. The veterinarian will also probably want to give some fluids, either subcutaneously or directly into the bloodstream. The method of delivery will ultimately be decided based upon the level of dehydration the vet detects. Champ will require a healthy, high quality large breed puppy food eventually. However, in the meantime, I would provide something like a Purina Critical Nutrition or HIll's A/D. These are prescription diets that are very high in fat and high calorie, and are typically not easily turned down by even the sickest of pets. Lastly, Champ probably needs your love, warmth and support. I am sure you have plenty of this.