The connection between the normal dimension and Crystal Dimension is not fully established and the only certain method of entry is the Crystal Gem Apple. In "Crystals Have Power" Finn gets into the dimension through a portal that a crystal sent by Tree Trunks opened. How the crystal that opened the portal got into the normal world of Ooo is unclear; Finn only says he "found this crystal on our doorstep."
Sometime before the series, the Rainicorns and dogs fought over at least some territory in the Crystal Dimension in the Rainicorn-Dog Wars, as stated by Jake in "Her Parents," but it is unknown if either of them succeeded. However, it is stated that Lady Rainicorn's parents Bob Rainicorn and Ethel Rainicorn were coming from the Crystal Dimension, so it can be inferred that the Rainicorns have some current stake in the area.