I think the Amber Spyglass is best for understanding Mrs Coulter's nature. In the books particularly Northern Lights, it's very much from Lyra's perspective where this glamourous figure steps into her life, utilises her then turns out to be the big baddie. The TV series has a more open perspective, so she is portrayed as showing some maternal moments, if conflicted. She is certainly determined and ruthless though and plays a very clever game in the patriachical world of the Magisterum. It's difficult to comment too much without spoiling, but one of the charms of the books is that the characters are realistic and complex. I can well imagine that suddenly on having a semi-independent child like Lyra into your life would stir up some unexpected maternal instincts.
The perspective is interesting with Will's background too and I'm really enjoying the way the detail of it has been brought in to overlap Lyra's timeline of Northern Lights. Will's viewpoint is much more of his mother's fears and obsessions appearing irrational, whereas we are seeing more substance to her fears being founded. The connections have been made much sooner than in the books, but that's probably necessary as otherwise it would feel like a seperate disjointed story that probably wouldn't work on TV. Books are more forgiving. Will and his mother are being represented brilliantly with justice being done to them.
My only criticism is the lack of armour in the battle. The book has a lot of detail about the gilt and pomp of Iofur's armour and the utilitarian function of Iorek's and how at the conclusion of the battle the rest of the bears cast aside their embellished armour and return to their true nature. The concluding move of the battle was Iorek charging up from feigning a wound, biting and ripping off Iofur's jaw. I can see why that wasn't included in full detail 
The critical detail is that a bear who is acting to his true nature can not be tricked. Lyra and Iorek discussed this in Trollesund. Because Iofur craved to be human, he ceased to be acting like a bear enabling Lyra and Iorek to trick him and win. They gave it a go at explaining, but I'm not sure how that would be picked up by people unfamiliar with the books.