I just climbed out of honor zero this week. I was at checkpoint three for about two weeks.
First advice: instead of typing your nasty messages, say them verbally. It will be just as effective in stress relief while your team will be none the wiser.
Be up front when you're new to something (such as a champion, position, or a matchup). My biggest problem was responding to flaming, so if you're up front, a good portion of the playerbase will take it easier on you.
Avoid jungle whenever you can. It is the least desired role with the most pressure on your performance. One mistake could piss them off, and we don't want that (as mentioned in my previous point).
Play support if you can and are good at it. If you can get your partner some good setups, you're likely to get at least one honor that game.
If you do get flamed, I won't tell you not to make a case for yourself. That said, don't obsess over it. Make your case without flaming them and move on. They may report you for inting, but even legitimate soft inters can avoid detection. You won't get in trouble for making a mistake here and there.
Most important: Don't give up! Take my advice, be patient, and do your best! If becoming honorable is what you want, it will come if you give it your all.