How does the choice between Kaathe and Frampt affect the game?

Having asked multiple questions, I'll try to answer broadly. The gameplay won't be affected by choosing Frampt or Kaathe. Kaathe encourages invasions, but you're still able to invade whenever you like. Your online experiences wouldn't be affected by this choice.

Siding with Frampt or Kaathe will not actually affect the ending; their dialogue may influence you in the choice made after defeating the final boss (and, having defeated them, you MUST make a choice and can't continue the current playthrough). The question of a "good" or "bad" ending in any entry in the Dark Souls series is purely up to interpretation. However, as far as access to areas and bosses, you're able to fight everything in the one run. Everything resets once New Game + (or NG+2, and so on) starts.

UPDATE: As mentioned in the comments to my answer, there are some pros and cons to siding with one or the other.

Kaathe allows you to join the Darkwraith covenant, which grants access to an item (the Red Eye Orb) which allows for infinite invasions. Frampt, on the other hand, allows you to "sell" items for souls, and he'll break down large titanite pieces into smaller ones. Siding with one causes the other to leave you for the rest of the playthrough. The Red Eye Orb and other Darkwraith-exclusive items will carry over to other playthroughs (however you will only be able to use the Red Eye Orb if you are a part of the Darkwraith Covenant).

The lack of these perks aren't insurmountable though. There are various occasions to farm souls in the game, and you can remain in the Darkwraith Covenant indefinitely, allowing you to use the Red Eye Orb, so long as you don't abandon the covenant.