She has been busy preparing for her upcoming trip to America.
Literal
She [topic-は] come-should America [toward-への] trip [possessive-の] preparation [object-を] do [purpose-のに] throughout busy.
来るべき ('upcoming, that is to come') uses ~べき in its attributive/literary sense of 'that which should/will come.' アメリカへの旅行 stacks へ+の to modify 旅行. するのにずっと忙しい means 'has been continuously busy in order to do.' Multiple modifiers create a dense but natural structure.