There are two common ways to add eyes to a head. First I'll describe how I do it when the eyes are meshes that are part of the head object:
One Way

Here is a head, consisting of 1 object. The object consists of two meshes, that I nickname half-a-head and left-eye:

Only half a head because I'm mirroring to save effort. They eyeball is a simple sphere that fits into a notch inset in the half-a-head and is rotated 90 degrees on X.
The armature for this part is three bones, although I only had to add and position two of them:
head
is positioned so that its head and tail are at the end point vertices of the half a sphere.
eye.L
is position so that it touches the end point vertex but the other end is at the center of the eyeball sphere.
The key is that eye.L
is parented to head
. I'll use Armature ▶ Symmetrize to generate eye.R
. This works because Blender knows about L/R or similar prefixes and suffixes to denote symmetry.
Next I select the head object and then the armature and parent with empty groups. Empty because automatic weight painting does a crappy job with this object. That creates a vertex group in the head object that has the same name of each bone:

Because the object is simple, rather than weight paint, I edit the head. First I select the eye.L
vertex group and the eyeball mesh and assign the vertices to the group. Then I select the half-a-head and assign it to head
. eye.R
takes care of itself.
Because the eye bones are parented to the head bone they move with the head bone in pose mode but can also be rotated independently of it.
Because the eyes are parented to the eye bones through the vertex groups, they move with the eye bones.
Once this is done, it's possible to add the rest of the eye rig without worrying about the eyes.
Here's the really simple demo file.

The other way
In this case, the eyeballs are separate objects, but the eye.L
and eye.R
bones are still part of the same armature as the head
and are still parented to it.
The difference here is that while instead of using vertex groups, we use bone parenting to parent each object to its relative bone.
The steps are the same for each bone, so I'll only describe eye.L
- Enter Object mode
- select the object
eye.L
- add the armature to the selection
- Enter pose mode
- select the bone
eye.L
- Parent and select Bone:

