I was recently critiquing an opening for a friend, because she'd said contest judges always noted that they didn't like her heroine. It was a puzzle, because she'd set the heroine up to be admirable-- a social worker who has organized a charity to help poor women get out of abusive situations.
What I realized is... sometimes deep POV in the opening of the story is counterproductive. If the character isn't easy to like, consider easing into forcing the reader to "be her".
In this story, the heroine was snarky. Really snarky. She had snarky mental criticisms about everything, about her boss's choice of footwear and the mayor's speeches, about her best friend's latest boyfriend and her own inability to get a date. Since we were in her head deeply and exclusively, we were sort of surrounded by all these snarky comments and couldn't get away.
So at first I tentatively suggested that the author just tone that down, make her less snarky. "But that's who she is," said the author. "And it's important that she be skeptical and critical because she's the only one who figures out that the congenial old mayor is a killer."
Okay. Well, you know, point of view "depth of penetration" can vary throughout the book, depending on what you need and how much you need the inner reality of the character. This is really important to know: Deep POV is not some life choice you make and can never unmake. It's just a tool to get what you want. And usually when you use deep POV, the purpose is to give the reader the experience of being this character.
Not all characters are good to "be" right off. Sometimes it might be better to ease into the character. You know how some people you don't much like right off, but as you go on you realize they're wonderful people, just gruff or curmudgeonly or sarcastic or whatever is offputting? (Interestingly, this can often make male characters more intriguing and appealing from the inside, but can make a woman character really hard to like. Sexism? Or is that just women readers' response? Maybe men don't respond so negatively to sarcastic women characters? Or maybe I'm the only one who responds that way?)
Well, sometimes it's better to present the character as she is on the outside, and hint at her inner depths, and then, when the reader already has reason to like her (because the outside draws him in), unleash the Seinfeld-within.
Just a thought. But you know, deep POV is not the only approach to introducing characters to readers. I bet you feel like you know Elizabeth Bennett and Mr. Darcy pretty well, and they were presented almost entirely from omniscient viewpoint. Sidney Carton? ("It's a far, far better thing I do...") Omniscient. All those books in the 19th century: If they weren't first-person, they were probably mostly in omniscient, and they were pretty good, right? You never put down Dickens and thought, "Boy, I don't understand that Scrooge fellow one bit. Wish I could be deeper into his mind."
You can get to know characters by their actions too. In fact, "By their fruits, ye shall know them," should be emblazoned on the computer screen of popular fiction writers. What the characters think and feel might be important. What they DO, however, is essential.
So if you feel like your character isn't immediately likable, but will be eventually, try easing in on the point of view. Start in single POV, but a little more distant. (I think that might also help us concentrate on an active opening, btw, rather than a few pages of snarky introspection. What's happening, not what's being thought?) Show the character through his/her actions and reactions, and just slide in thought and feeling as needed.
You're in charge here. Never forget that. You are not controlled by your POV choice!