His insisting on claiming his supposed title is nothing more than meaningless posturing on his part, as he wasn't Lord of Harrenhal anymore by the time he arrived at The Wall. He was stripped of that title by Tyrion, and Harrenhal was subsequently given to Littlefinger.
As far as his influence on the wall, I don't think it's even made explicit, but there are a number of things that Slynt has going for him over your average raw recruit:
- He was a former Commander of the City Watch (the Goldcloaks), and Lord of Harrenhal, so he would probably get a higher starting position. Remember our very first scene in A Game of Thrones has a former nobleman, now in the Night's Watch, outranking a clearly more experienced and qualified Night's Watchman (before getting Other'd to death).
- He rapidly allies himself with existing ranking members of the Watch, like Thorne. By accusing someone that many of the brothers already disliked, he earned their allegiance; and 90% of getting a position of power is simply having other people accept that you deserve that power.
- He has the explicit backing of the Lannisters, both Tywin (who's filthy rich) and Joffrey (who, by that point, had basically won the war.) Again, that makes it likely that other ranking Watch officers would follow his lead, in hopes of earning favor.
Of course, a subsequent events will ultimately demonstrate, he has no official authority in the Night's Watch, only that he can derive from his high-powered friends:
He gets no support from Stannis nor most of the Watch when he makes a play for Lord Commander, and ultimately, Jon tries to send him away. With Tywin no longer alive to support Slyn, Jon simply executes him for insubordiation when he refuses.
