That's the thing. Nobody knows exactly how it works. The honor system is ambiguous.
The best we can tell you is to be positive and don't get "legitimately" reported, not that you get any indication of that though. There's really no way for you to determine what you're doing right and what you're doing wrong. Ultimately it is just a grind, and one that needs to be started early.
I don't think you have a shot at getting honor level 2 this season unless you're on your last checkpoint. The season ends November 12th and right now people who are on the 3rd check point have a very slim chance of getting to honor level 2.
Riot should tell people that. Before if you got punished after a certain date, then no rewards for you. The system was clear which is fair. Now they're forcing people to grind in the hopes that they'll reach honor level 2 when we have no way of tracking our progress and the checkpoints are so long. That's the messed up part.
Personally I think you should get ranked borders regardless, but that will be locked for anyone below honor level 2 along with the skin.
If you want a reference point, even though for everyone it is different, I've played approximately 220 summoner's rift games, and around 10 aram games. I was bumped back to honor level zero on 7/21. Last week I just got to the third check point of honor level 1. I don't expect to unlock honor level 2 for at least 50 more games and I'm unsure if I can even spam that many with midterms happening.
It is different for everyone of course, but you're looking at 100 summoner's rift games easily. It may even be much more than that, but I haven't seen anyone report lower than 160 games to get from honor level 0 to honor level 2. The duration of the game is taken into account too, so spamming ARAM games doesn't work.