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Hello there, I'm studying to be a Japanese Teacher at university (I'm on the later half of my second year) and I often study by myself as well (which has helped me a lot in the proficiency examinations), so I guess I know how to help you here.
I'm not sure about what you mean when you say you learned "75" kanji, but I believe you learned the meaning and possibly one or two readings. This would sure help you in the future and work as a mnemonic exercise, but I strongly advise you not to do it as it might burn you out unnecessarily. Not to mention that, if you begin with kanji, you won't find much use to them, so you're bound to forget as fast as you learn. Besides, you won't learn the tones that way.
Floorfiller's order was pretty accurate, but I'd like to write a bit more about it 
- Hiragana: Best place to start. In the beginning, run away from romaji like the plague, as it may lead to problems with your pronounciation and you might also get addicted to that "easier way of writing". Pay proper attention to stroke order as well. As for katakana, do learn it but don't mind it too much - it's not nearly as used as hiragana (or kanji, for comparison basis) so obviously it will take longer to get used to.
- Grammar/vocabulary: Naturally, you need to improve step by step. Learn the daily expressions, ways of introducing yourself and, above all things: listen to the language as much as you can, specially because the pronounciation of the Japanese language seems to be a bit tricky to people whose mother tongue is English (specially the intonation). You're going to get used to hiragana and you will slowly learn the kanji along with the vocabulary, so there's no need to rush.
Learning how the tones work is quite interesting if you want your Japanese to sound natural. Most likely, you're listening to the language spoken in Tokyo, but tones tend to change more than anything in the various dialects so watch out not to get yourself confused.
- Kanji: Think of kanji like numbers. They give you an idea and a hint about what the pronounciation might be. The same way "2" can be read as "two" and "20" can be read as twenty and "2nd" means "second", "今" is read "ima" and "今日" is read "Kyou" and "今年" is read "kotoshi". The idea of "now" is somehow present, as the sense of "pair" exists on "2". Do not, by any means, learn too many kanji with once in the beginning. Chances are you're overlooking stroke order and the importance (or existence) of radicals, and that will cause you problems in the long run.
- More grammar/vocabulary: Language is like a board game. Grammar are the rules and vocabulary are the pieces. If you have too many pieces and you don't know the rules, the pieces are quite useless. If you know all the rules, but you don't have enough pieces, you won't go too far, but at least you know what you're doing. Therefore, give grammar a special attention, and try to use the vocabulary you know as often as you can. Don't try to master Japanese in a heartbeat. Repeat, repeat and... yes, repeat. Keep practising, both the written and the spoken language. Also try to listen to the language as much as you can.
As for the difficulty of the language, it is on the eye of the beholder. Sure the structure is different, but it is so awfully simple that young Japanese kids often use structures that foreigners learn after years and years of study. Besides, after you practise for a while, you get used to it. The language is also often so regular it isn't even funny. I'd say the hardest part is indeed the writing (having those bloody kanjis all over the place, with several different readings). Other than that, there are the natural difficulties of every language: acquiring grammar and vocabulary, to the point you become fluent.
So, answering your question: Can you learn Japanese by yourself? Sure you can. All teachers do is showing the path, you do the rest. However, if you're not used to learning languages, having the help of a teacher is strongly advised. Personally, I study by myself as much as I can, getting all sorts of extra info I can, but I often ask my teachers for more material/learning strategies 
ps.: Get yourself a proper book (not those pocket ones), a couple of dictionaries (English <-> Japanese; kanji dictionary) or similar websites. The books/on-line courses should provide you a good methodology and the dictionaries, well, they're always handy.
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