I have been in the fanfiction reading game for almost three years. My experience in reading fanfic tions is along with the professionals. So I am one to be recommended for in the Beta-Reading process.
I look for two things in good fanfics. One that gives a good story, and one that gives good grammar. If one fails, then both fail. I won't tolerate "ur" for "your" or "tht" for "that". So anyone requesting my help, be sure to undo those spelling shortens.
I read fanfics justly and unbiastly. If I see a good story, I won't judge it as bad because of the pairing favouritism. Likewise, if I see one of my favourite pairings requested, I will still turn around and say it was pathetic because of the poor storyline.
I have a keen eye for good stories, so if you're really confident in your works, come to me, and I'll read it without hesitation.
Which series I will read:
Yu-Gi-Oh (Only part 1, I'll consider part 2, but not anything over that)
Sword Art Online
Final Fantasy I-X (Nothing beyond X)
Jojo's Bizarre Adventure (Up to Stardust Crusaders, not seen any more than that)
Game of Thrones
Lord of the Rings
Super Smash Bros series
Legend of Zelda
Attack on Titan
If there's anything here which you've not seen, gimme a wee shout :)