The phrase "on-screen" pretty much always means "something dsiplayed on the screen of a computer" (or in some cases a television or movie screen). Metaphorically, it can mean "directly visible" as in

The character Jones never kills anyone on-screen, but we learn of several off-screen killings.

This form could be used even of a novel, where of course nothign is truly displayed on the screen, because there is no screen. "on-stage" can be used in a simialor metaphorical way.

But if you mean:

Her eyes were looking intently at the computer screen.

Then "on-screen" should not be used, as her eyes are never displayed on the screen Instead, you could say:

• Her eyes would be focused on the screen.
• She was staring intently at the screen.
• Her gze seemed locked on the screen

The phrase "on-screen" pretty much always means "something dsiplayed on the screen of a computer" (or in some cases a television or movie screen). Metaphorically, it can mean "directly visible" as in

The character Jones never kills anyone on-screen, but we learn of several off-screen killings.

This form could be used even of a novel, where of course nothign is truly displayed on the screen, because there is no screen. "on-stage" can be used in a simialor metaphorical way.

But if you mean:

Her eyes were looking intently at the computer screen.

Then "on-screen" should not be used, as her eyes are never displayed on the screen Instead, you could say:

• Her eyes would be focused on the screen.
• She was staring intently at the screen.
• Her gaze seemed locked on the screen
• Her eyes would be set on the screen.
• Her eyes would be fixed on the screen.

or any of several other possibilities. All the above have very simialr meanings.