How to gain influence:
First rule is simple: Talk.
This can't be stressed enough. I can't tell you how many times I've appointed character A for a position, and had character B complain later (either to the realm, to me directly, or to someone else who passed it on to me) that they had more experience and had been in the realm longer, so they should have gotten the position.
But character A spoke up frequently, while character B just quietly followed orders.
This is also the reason why, despite what Bedwyr said, the value of going on IRC cannot be discounted. There were plenty of times where character A had also been relatively quiet in-game, but had come on IRC and chatted quite a bit—and done so in such a way as to show his or her intelligence and/or trustworthiness.
Is this entirely fair, using OOC knowledge of their intelligence and trustworthiness to make the decision? No, it's not. But it's also unavoidable.
I would also say that the way of talking that's most likely to get you positions (in my view) is to
wait a while after you've joined the realm, then start making suggestions. And I'm not talking about, "Y'know, I really think this realm would do better if it were a democracy instead of a monarchy" kind of suggestions: I'm talking about, "Y'know, we could really use a few extra nobles scouting that border" or "I've got some ideas for how we could make our food distribution more efficient" type of suggestions.
It's also hard to stress enough that it's not sufficient to talk, and to make suggestions. If your talk is annoying, or your suggestions are either given in an irritating manner (for instance, sounding as if you think you're smarter than the Council), or are
bad suggestions, you're not likely to get very far. There are a number of people I've dealt with—some both in-game and on IRC—who have been nothing but nuisances, from failing to understand why I didn't give them positions, when they were
clearly the most qualified around, to failing to grasp the basic principles of how a monarchy is supposed to operate (hint: it doesn't usually involve a lot of voting).
So: yes, talk, but don't think that the mere act of talking alone is sufficient to get you positions.