Can I transfer my Xbox 360 copy of Grand Theft Auto V to Xbox One?

Can I exchange or trade my Xbox 360 version of Grand Theft Auto V for the Xbox One version?

Currently, Xbox has no direct way to do that. If you have a physical copy, you likely have a local game retailer that accepts trade ins. Trading it in for the Xbox One copy is effectively the closest you will get. If you have heard of others being able to do this, they likely play on PlayStation, which had a similar promotion around the release of the PlayStation 4.

Can I play my Xbox 360 version of Grand Theft Auto V on Xbox One?

No. At least, not yet. Xbox One now has backwards compatibility, allowing Xbox 360 games to run on the console. However, each game needs to be specifically enabled, due to the way the emulation software works. Grand Theft Auto V is currently not on the supported list, but backwards compatibility is still in early stages, and Microsoft has promised the eventual support for hundreds of titles.

To quote Microsoft;

"Xbox One Backward Compatibility is here. Now you can play a growing number of your Xbox 360 games on Xbox One at no additional cost. With over 100 titles slated for this fall, and hundreds more in the months to come"

Allowing speculation, I find it less likely they would support a game already on the Xbox One. Their main focus appears to be earlier games that would otherwise be unplayable on the new generation.

Can I transfer my Grand Theft Auto V character from Xbox 360 to Xbox One

yes, you can. Grand Theft Auto V is one of many games to support transferring saves from old to newer generation consoles. In fact, doing so unlocks exclusive content, as a "returning player" bonus. The service is free, and you can find more information regarding how process the transfer here.