Surprisingly, I'm not really that obsessive at all over characters that I like.
When I watch something and find such a character I'll usually:
a) buy a figurine (though I'm quite fussy about the face and quality, if it isn't good enough I won't buy it)
b) look up fanfiction about said character in a pairing
c) sometimes I'll look through official art or fanart
d) rarely, I will actually try to do some fanart or fanfics of my own
For Sesshoumaru I have a wallscroll too, but that was kindof a random buy. If I had more space then I'd likely buy more wallscrolls of a few other characters that I like.
And that's about it