Selecting which claims to check is a time-consuming task for human fact-checkers, especially from documents consisting of multiple sentences and containing multiple claims. However, existing claim extraction approaches focus more on identifying and extracting claims from individual sentences, e.g., identifying whether a sentence contains a claim or the exact boundaries of the claim within a sentence. In this paper, we propose a method for document-level claim extraction for fact-checking, which aims to extract check-worthy claims from documents and decontextualise them so that they can be understood out of context. Specifically, we first recast claim extraction as extractive summarization in order to identify central sentences from documents, then rewrite them to include necessary context from the originating document through sentence decontextualisation. Evaluation with both automatic metrics and a fact-checking professional shows that our method is able to extract check-worthy claims from documents more accurately than previous work, while also improving evidence retrieval.