I think she’s related to the Apple clan by her father marrying in. Her father runs a huge carrot farm in another town, and married a cousin of Apple Jack’s late mother.
 
They had three kids - Carrot Cake, Carrot Top (known to the Apple family as Golden Harvest, or just “Goldie”) and Noi - in that order. Carrot Cake was the first to move to Ponyville, to start the Sugar Cube Corner bake shop.
 
Later, after the death of Apple Jack’s parents, to alleviate the resulting financial hardships, the Apple Family sold an unused parcel of land nearby to Carrot Top’s parents, who then sets up Carrot Top’s own carrot farm there.
 
Whenever Carrot Top’s parents went on vacation, they would have Carrot Top foal-sit her little sister Noi (much like how Rarity takes care of her little sister Sweetie Belle).
 
Later, Derpy, a struggling mailmare and single parent, would move in with her own foal Dinky, using her rent to help Carrot Top with the expenses.
 
In return, Carrot Top also takes care of Dinky when Derpy is at work. Because of this, I figure that Dinky would refer to her as “Aunt Goldie”. Also, Noi and Dinky would become best friends and almost like sisters.
 
Well, that’s my head canon…

@Cirrus Light  
I’m working on a comic how Carrot Top get Applejack as a pet

@Background Pony #E9EF  
Buuuut that comedic conjecture was in ignorance of the source :p

@Background Pony #E9EF  
I was going to chalk this up to Golden buying out AJ’s farm and giving AJ the chance to keep it by accepting a Rickshaw job from her, partially to assert her dominance / victory over the Apples :p
 
But, her being an “Apple” would make that less likely.
 
Unless she’s really even more of a jerk, lol.

I’d just like to point out that Golden Harvest shows up in Apple Family Reunion, so she looks to actually be related to Applejack.
 
On top of that, judging by her showing up in 3 or 4 episodes with the 2 same ponies (Fire Chief and an unnamed stallion), she seems to be married with a foal. And Fire Chief shows up with Amethyst Star in Sisterhooves Social… Which might mean Amethyst Star is Golden Harvest’s daughter too. So Golden Harvest is, by the looks of it, married, a lot older than AJ, and related to her.