How do you go about doing it? Especially on your own?
Well, the best way is a combination of self-teaching, formal classes, and a native Japanese speaker to practice active conversations with. Well, the BEST way is to live in Japan for a while, but the above is the best way without actually leaving your home country.
Many of us do not have access to formal classes or a native Japanese speaker. If you don't have access to these, you'll have to rely on books and imported entertainment (like video games!) to help you learn Japanese.
For grammar, there are plenty of books that will teach you good Japanese grammar. The same is true of vocabulary. Where I had the biggest problem, personally, was learning Kanji, those really complicated pictograph-style characters.
The best book that I have found for teaching Kanji is Remembering the Kanji by Heisig. Well, it's actually a three part series of books. The first book teaches you the meanings of around 2000 Kanji... and you don't even need to know Japanese to learn them. The theory is that it's MUCH easier to remember all of them if you learn the meanings and the pronunciations separately. The second volume then teaches you the pronunciations of the same 2000 Kanji. The third book adds another 1000 less-used Kanji or so and teaches you the meaning and the pronunciations, again, separately.
On top of that, there is one tool that is almost indispensable when practicing your Japanese in video games, and that's an electronic Kanji dictionary. The cheapest way to get one that recognizes handwriting (very important for finding characters you are unfamiliar with!) is to get Kanji Sonomama Rakubiki Jiten for the DS. It is sometimes called Kanji DS Dictionary, and it is exactly that, an electronic Kanji dictionary for the DS. It's worth every penny, especially when you consider that dedicated Kanji dictionaries can cost several hundred dollars.
To buy Remembering the Kanji: Volume 1, try HERE
To buy Remembering the Kanji: Volume 2, try HERE
To buy Remembering the Kanji: Volume 3, try HERE
To buy Kanji Sonomama Rakubiki Jiten, try HERE
Have fun learning!