The "across" version is quite unusual, and the "throughout" version sounds a bit stilted. The most common way of expressing this (I'm imagining an annual performance review):
I've accomplished a lot this year.
If the context doesn't make clear that this refers to the past twelve months, one could also say
I've accomplished a lot this past year.
My answer holds for US English -- I can't vouch for other flavors.
I checked the link you provided in a comment. The speaker's context is quite different from "I accomplished a lot [preposition] [time period]." In the TED talk, the speaker was talking about the calendar, the months of the year. "Across the year" makes sense in that context. We can see the months ticking by as we go along the timeline. In that context, "across" is probably better than "throughout," because "throughout" suggests that something is constant, or fairly constant, for all 365 days.