I'm not sure exactly how would be best to help you, but I suppose the least I can do is give some tips regarding gaining Honor;
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Don't spread negativity. This is a seemingly simple and easy thing to do, but it can be surprisingly easy to fall into the habit of making defeatist or passive-aggressive remarks when things go south in League. Negativity like that can validate a report, and valid reports slow your Honor progress, so it's worth pointing out that one should always follow the golden rule; "If you don't have anything nice to say, don't say anything at all".
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Be positive as frequently as you can manage. This one is definitely speculative, but regardless of whether or not it actually works, it can't hurt to put your best foot forward. It may or may not influence your Honor gains (since, if you lose/slow down in progression for misbehavior, it only makes sense to get increased progression for good behavior, right?), but positivity can definitely help influence the likelihood that your teammates will Honor you, and every small boost helps, as it were.
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Randoms > Premades. Premade Honors, as it were, count for less than those commendations offered to you by teammates from without your social circle. While I'm not sure exactly how great a difference it makes to be Honored by randoms over premades (or, for that matter, if premade Honors even give any progress), it is still a fact that Riot designed the Honor System to prevent Premades from exclusively Honoring each other to optimize their benefit.
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Always Honor someone. While you'll always gain a modest amount of Honor progression from simply playing matches without validating reports, and you'll get a small boost if you yourself are Honored, the biggest factor in Honor progression is the Team Honor Bonus, when all members of your team Honor one another. So, it's worthwhile to always try and Honor at least someone, just to contribute to the Team Honor Bonus. Not mandatory by any means, but it helps.
Those are (in the case of 1, 3, & 4) some facts about the system, as well as (in the case of 2) a speculative tip to help in the Honor climb.
Without any further information, it's hard to really say much else - partially because, like with the IFS, a decent amount of understanding the Honor System is speculative with no hard data, just anecdotes of player progression. That said, if you've any further questions, I'm absolutely glad to answer as best as I'm able, and I hope that what information I've given already can be of help to you.
