David Tennant is a much darker character than he often is in other works- dour, brusk, uninterested in peoples' feelings, and intensely private. He is also sick with a heart condition and looks it: he's frequently sallow, dishevelled and hollow-eyed, tired and ailing. Miller is a good-hearted person who would be more than willing to meet Hardy half-way if he made any effort to be personable (he generally doesn't). She's also a good cop, though inexperienced with murder investigations, and is no pushover, holding her own with Hardy when he impatiently tries to brush off both Miller and her concerns. This case is especially hard for her because she's from the community: her son was friends with the victim, and she knows all the involved parties including the suspects.
The acting in Broadchurch is superb: not just from those in the main roles, but from everyone even in very minor parts. The portrayals of the grief and guilt, anger and pain felt by the family of the murdered boy- and the community at large- ring true and are very effective. The plot is absorbing and keeps you guessing as to who might be responsible; I figured it out in the second to last episode due to something said by DS Miller, but for a good deal of the series, it's impossible to guess the perpetrator with the available information from the investigation. All in all it is a really well done and absorbing show, and I'm looking forward to watching series II.