Hi Mia,
From your attached images, it looks like you only have one unit cell displayed. The carbon atoms outside the unit cell only appear because they're bonded to atoms within the cell and you've checked the default 'Search atoms if A1 is included in the boundary' box in the 'Edit | Bonds...' dialogue. While this draws any extra atoms required to complete the bonds, it doesn't extend the volumetric data in the same way. Try going to the 'Edit | Bonds...' dialogue and chose 'Do not search atoms beyond the boundary' and you'll see where the actual boundary of your structure is.
If you do wish to extend both the atoms and volumetric data, the correct way is to click on the 'Boundary...' button at the bottom left of the screen and increase the ranges of coordinates. To reproduce your figure CHGCAR_1LiC6_SCF_a, for example, try plotting over the range -0.5<x<1.5, -0.5<y<1.5, 0<z<1. If you want a boundary which is not parallel to the (100), (010) and (001) planes of the structure, you can use the bottom half of the 'Boundary...' dialogue to create additional cutoff planes.
The procedure for visualising just one Li atom and its coordinating C atoms is exactly that: adjust your boundary, using the range and cutoff planes as necessary, so that only the required atoms are included within the boundary. In CHGCAR_1LiC6_SCF_a, I think the appropriate range is -0.5<x<0.5, -0.5<y<0.5, 0<z<1 with no cutoff planes required, but you'll need to check that as I don't have your VESTA files to play with!
I've assumed that all you want to see is those thirteen atoms, isolated from the structure. It is also possible to show the volumetric data for just one set of atoms, but in context, surrounded by the rest of the crystal structure. To do this, you'll need to create a second phase with exactly the same unit cell and atomic structure and position it with a displacement of (0,0,0) relative to the first phase. Now reduce the boundary of the phase with the volumetric data and leave the identical second phase in place to display more atoms.
I hope that helps; do let me know if you need me to explain any of the above steps further.
Philip