Why didn't Doug kill Rachel from the beginning?

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Doug had feelings for her that he couldn't comprehend or deal with. His is a profoundly lonely existence, that loneliness caused his own mind to rebel against him, forcing him to take grave risks. It's a risk of power. The more power you hold over others, the less you can relate to them, and by extension, yourself. If you don't take active steps to alleviate that loneliness, then it will bite you when you least expect it.

Doug's entire character arc in the third season was devoted to resolving this key character deficiency. Doug's smart and self-aware enough to realize he has this problem, and reaches out to his brother for help, not because he really needed help physically, but because he needed to feel that human connection again. Of course, once he got what he needed from his brother, he summarily booted him back out of his life.

So when he finally tracks Rachel down and has to choose between killing her or letting her live, the loneliness didn't drive him towards the wrong decision.

One of the key differences between Frank and Doug that underlie their working relationship is that Frank has learned how to let others help him, whereas Doug has to do everything himself. It's why Doug is *just* a chief of staff and not terribly involved in the political process, though that's started to change in the fourth season, you can see his growing pains in his interactions with Seth.