In Hoard of the Dragon Queen: Episode 1, one of the encounters is a dual between one PC and a named NPC. This encounter is supposed to be quite deadly, a level 1 or 2 PC versus a CR 4 NPC. My players have level 7 characters, so I am adapting this for their power and a CR 4 NPC is a "hard" encounter for a level 7 PC according to the Kobold Fight Club encounter builder. How can I make the CR 4 NPC appropriately deadly for a level 7 PC?
The duel isn't meant to be deadly, it's meant to be impossible. A CR 4 creature would be a deadly encounter for a single level 5 PC.
With that in mind, you need to know that your players fight this NPC again towards the end of the 3rd chapter. At that time, they would usually be 4 level 3 PCs, so a CR 4 creature would be a hard encounter. He has other NPCs with him for that fight that brings it up to a deadly encounter, but that's not important for your current purposes. I'm going to assume that you're not planning on having them level up between now and this fight.
So your goal here is for him to be unbeatable for a single level 7 PC, and a hard encounter for 4 level 7 PCs. A hard encounter for 4 level 7 PCs ranges from 4400 to 6800 XP. Given that you want him to be as difficult as possible for the duel, he should probably be at the upper end of that. A CR 10 creature is worth 5900 XP, and a CR 11 creature is worth 7200, so I would try for at least CR 10, possibly CR 11.
So now that we have our target, we need to get him there. He's currently a CR 4 creature with the hit points of a level 6 Barbarian, the class features of a level 5 Champion Fighter, and the proficiency bonus of a level 4 character. In keeping with that, I'd probably give him the hit points of a level 11 Barbarian, the class features of a level 10 Champion Fighter, and the proficiency bonus of a level 9 character.
This should result in an NPC who can brutally crush a single level 7 character but can be defeated by 4 of them in a tough battle.
The easiest and most painless way to increase the CR of an encounter is to give the creature one or more extra Hit Dice. The creature itself will have the same power, as you aren't boosting its attributes, attack bonuses, etc. It will just be slightly harder to kill it.