We normally use the, the definite article in English, to indicate that we are talking about a specific object. He was looking at the coils of line because he was looking at specific coils of line - those which were on his boat. They are just coils of line, though, because line in this case is just a generic material. It is not any specific line, just line. (This absence of either a definite or indefinite article is called zero-marking or the zero article. You will also see this phrasing in sentences like "pieces of pie" or "slices of cake".) The ocean is similar to the first sentence: it is the specific ocean that he was in.

Here is a good article on definite and indefinite articles from the Purdue Online Writing Lab: https://owl.english.purdue.edu/owl/resource/540/01/

Here is an article on the zero article: http://www.bbc.co.uk/worldservice/learningenglish/radio/specials/1113_gramchallenge30/

To breaking point is a little more ambiguous. I would ordinarily say to the breaking point (after all, it is a specific point) but simply to breaking point is idiomatic too. Here is a graph of the usage of both "to breaking point" and "to the breaking" point, which shows that both are in use.