In his book, Staring at the Sun: Overcoming the Terror of Death, existential psychotherapist Irvin Yalom argues that to overcome the fear of death, people must stop avoiding it.
Yalom writes that earlier in his life, he was sometimes overcome with death anxiety. This (perhaps) contributed to him becoming a psychotherapist, and in particular, to focus on existential issues. Yalom draws on several decades of clinical experience (as a psychotherapist) with people experiencing death anxiety in writing Staring at the Sun.
Yalom's main point is that most cultures (though less so in Eastern cultures) do not promote an awareness of death. In turn, something that is naturally fear inducing (death) becomes even more scary than it would otherwise be. It is sort of like that task (or tasks) we've all had in our lives that we know we have to deal with, but we keep putting it off and putting it off. And, in doing so, we become more fearful and anxious about doing it. The short term fear/anxiety it will cause prevents us from dealing with it. But, if we did the task, we probably would feel a lot, lot better. By ignoring it and forcing it out of our mind, we intensify the anxiety it causes.
Because death, like the task we put off, will always enter our minds from time to time, we aren't avoiding the anxiety entirely. But, by pushing death out of our minds, we do avoid thinking about it for now. And that is the key, for Yalom, that intensifies death fear. Put differently, short term death anxiety forces a pushing of death thoughts out of one's mind, but it actually increases death anxiety over time.
For Yalom then, to truly face one's death with full conscious thought and feeling is to overcome death anxiety. Of course, this, at first, is not entirely pleasant (especially for some of his patients who are likely to fear death more than the normal population). But, when we face death head on, it becomes less of a monster in our closet, so to speak. It doesn't become wonderful or anything, but it becomes ok.
There is some empirical data that supports these ideas. Research I've headed (unpublished for now) shows that in the short term, reminders of death increase depression. However, thinking about death for 5 minutes a day for a week actually reduced depressive symptoms one week later (compared to thinking about uncertainty or a control topic).
It also is (roughly) consistent with research on post-traumatic growth. This research shows that not only do most people recover from trauma fairly well, but that they often report experiencing a lot of growth from the experience. For instance, many cancer patients report that when diagnosed, although terrifying and numbing at first, they begin to see life completely differently, almost as if they are living fullly for the first time.
So how do we overcome the fear of death? The key is to stop putting off and avoiding the topic. It isn't like we are going to not die, so avoiding death thoughts can never be a successful strategy.