I managed to teach myself Koine a decade ago using Hewett's New Testament Greek: A Beginning and Intermediate Grammar. It was, not surprisingly, a hard slog, but by the end I could make my own way through the gospels or Revelations fairly confidently. Then, after neglecting my Greek for years, I decided to brush up on it with a different textbook, since I had written all the solutions to Hewett's exercises directly into that textbook.
Vine was easy to get and cheap, so I figured why not. Unfortunately, I am sceptical that even the most motivated autodidact could learn Koine with only Vine's text. Perhaps if you already knew Latin or another heavily inflected language, and also had a bit of a masochistic streak in you. Vine's method gives you a huge dose of nouns and adjectives and pronouns and declension right at the start - the first 5 chapters give no verbs other than 'to be', in fact. For native speakers of English this is terribly tough to get through, since English is such a verb-centered language. Then, when verbs are finally introduced in lesson 6, Vine gives you all 6 tenses of the indicative mood at once. WOW. I am a professional linguist and that scares the begeepers right out of me. The whole learning process seemed pretty smooth when I was using Hewett's method, which is much more verb-centered, but I shudder to think what it would have been like if my first taste of Greek had come through Vine.
Vine chose to exclusively use verses from the New Testament to form the exercises for his method. This is a noble thought, but it means that towards the beginning of the book, he must help you with direct translations of multiple words from some sentences. These translated words, which you don't know yet at that point, would have distracted me from learning the core material of the given chapter. This method also means that Vine can dispense with writing a key to the exercises, since he can just point you to the NT verse in question, but your Greek will then be colored by whatever English translation you are using to check your work. I am not sure that this doesn't defeat the whole point of learning Koine in the first place, which is presumably to free you from other people's translations and the inevitable interpretations that they carry with them. I preferred Hewett's approach, where he invented the sentences in the exercises at the beginning of his textbook and provided a key to go with them, and then started to introduce some complete NT verses towards the middle of the book.
There are also several annoying details about the edition of Vine that I have (1997, Thomas Nelson Inc.). The breathing marks above letters are so small that I sometimes feel I need a magnifying glass to determine if the mark is hard or soft, and if you try to use this book anywhere but at home at your desk, you may not be able to make them out. I suppose that is really the fault of the font selected, but in that case a larger font could have been chosen. In addition, some of the intratextual references are wrong, e.g. a reference to what is on p. 48 leads you to something different from what was intended; the correct reference should have been to p. 28. Those errors must have crept in when this work went through a new edition at some point.
I did fine with Hewett, and I have heard good things about Mounce as well, but I recommend you not try to save money by using Vine.