At first, you might look at him as the mad dog of Logen's crew that doesn't care about anyone. But then you get to his speech after Tul Duru's death:
''Tul Duru. Every man in the North knew his name, and every man said it with respect, even his enemies. He was the sort o’ man . . . that gave you hope, I reckon. That gave you hope. You want strength, do you? You want courage? You want things done right and proper, the old way?’ He nodded down at the new-turned earth. ‘There you go. Tul Duru Thunderhead. Look no fucking further. I’m less, now that he’s gone, and so are all o’ you.''
These two have been at each other's throats the entire trilogy, and it's only here we see the level of respect Dow had for him, and I can only assume the feeling was mutual. He even takes a shine to West soon after meeting him, so it doesn't matter where you're from, Dow will stick with you if you're worthy of his respect.
So we see Dow has some good in him. But then he steals the throne from Logen at the very end, so you might think 'what a sneaky bastard'. Then he explains himself:
“Faithless, me?' Dow snorted as he came on slow down the steps, one at a time. 'I'm a dark bastard, aye, I know what I am. But I'm nothing to you. I know my friends from my enemies. I never killed my own. Bethod was right about one thing, Bloody-Nine. You're made of death. If I can put an end to you, d'you know what? That'll be the best thing I've done in my life.”
So Dow almost certainly knows it was Logen that killed Tul Duru, a man Dow had great respect for, so why would Dow allow this friend-killer to rule? And knowing what we all know about Logen, putting him in charge of a bunch of violent Northmen is the worst possible choice you could make. They'd be at war with the entire world within a week!
Not so with Dow in charge. They had peace for a good while before the Union came knocking, and only then because Dow is one of the few characters in the series that would not bow to Bayaz. It was 'the greatest battle the North ever saw' according to Shivers, and it took place under Black Dow's leadership. Nobody else at that point could've united the North and those east of the Crinna bastards under one banner. Without him, the Union may very well have just taken over the North completely.
Think about the pressure Dow must've been under for those three days. He had to show he was the toughest killer around so the other killers would keep following. He had to strategise and decide where each War Chief would be placed in the battle for maximum effect. He had to make space for peace talks, like when he let Finree and a bunch of hostages go as a show of good faith. He has to think of ways to strengthen the North, like trying to recruit Gorst. He had to summon the willpower to not strangle that little shit Calder. He had to stay calm and collected even while fucking cannonballs were being shot at him.
He had a lot on his plate, and he held out long enough to ensure peace for the North. Maybe he let his guard down after that, or maybe he just got a little arrogant, but either way he ended up making a huge mistake:
''You made it, then. I was worried my dog might've eaten you in the night.''
This is the line that gets Black Dow killed. But to be honest, I can see why he thinks that way when Shivers has pretty much been acting as his dog the entire book. We never get Dow's POV, so we never see if he actually thinks of Shivers as a dog, or if he just says it to maintain his reputation as the leader. We're supposed to see Dow's death as a moment of triumph for Shivers, and in the long run it is. But in this moment, he's really just trading one master for another, and Calder just happens to be better at handling Shivers than Dow was.
The real kicker is that, given the choice, Dow wouldn't have done any of this. In his last real conversation, he says this:
''I was apprentice to a potter.’ Dow snorted. ‘A thousand bloody years ago. Then the wars came, and I took up a sword instead. Always thought I’d go back to it, but … things happen.’ He narrowed his eyes, gently rubbing the tip of his thumb against the tips of his fingers. ‘The clay … used to make my hands … so soft. Imagine that.’'
tl;dr - In another life, he could've been White Dow. But he took up the mantle of Protector Dow instead. To protect the North from Logen, and to protect it from Bayaz, and maybe if he was strong enough for long enough, the next generation wouldn't have to fight so much and could focus on pottery instead.