First of all, are you sure DK is part of the background? I'd be surprised if he wasn't a sprite. You may want to read about how the NES PPU works:https://wiki.nesdev.com/w/index.php/PPU
The problem you or anyone else will have with editing Donkey Kong is that it's a very small ROM with very little space, though I'm thinking of the, er, original version, rather than the Original Edition. Maybe the latter has been expanded, leaving more empty space.
Replacing the tiles is the easy part, but if you want to change the animation, you'll need to learn assembly. There will be code in the game saying "use this tile here" for each part of DK, so to change where the tiles go, you'll have to edit the existing code, and possibly add more somehow. I'll have a look myself later.
In the meantime I recommend learning about how the background and sprite layers work, and remember that pixels don't mean too much on the NES because the system works entirely on 8x8 tiles. Actually this is true for most 8- and 16-bit consoles.
EDIT: you were right: DK is part of the background rather than a sprite, at least most of the time. When he carries Pauline up the ladder, he becomes a sprite. Around $330 in RAM is where the tile arrangement is kept for DK, and it's sent to the PPU from there. It gets the details from around $4668 in the ROM. You'll find that $4668 to $467F is the tiles used for DK's first frame, 4x6 tiles going in columns from left to right. The previous address ($4667) contains $46, and I'm not sure precisely what that does, but the point is that you'll find 24 bytes with one frame of DK, then a $46, then another 24 bytes with a different frame. Try going to $4668 in the ROM and start messing around with it!
EDIT2: In case you didn't guess, $24 is the tile used for the black background.