ゴミ箱がゴミで溢れているって英語でなんて言うの？

"The trash can is completely full and overflowing."

As stated, this expression implies that the trash can is completely full of garbage or trash and trash is spilling out.

"The rubbish bin is completely overflowing."

This is the British English for the previous expression "the trash can is overflowing."

"The trash bin is overflowing with garbage."

This is a slightly more formal and detailed way to express that the garbage can is overflowing with trash.

Note: The British use a different word for trash or garbage than Americans. The British usually say "rubbish" instead of "trash" or "garbage."

An overflowing trashcan.

We use 'an' because the following word begins with a vowel sound (o).
'Overflowing' - refers to the bin being filled with trash to the point that the trash is spilling out.

Example
A: Isn't that gross?
B: What?
A: That overflowing trashcan.

A: You need to take the trash out now.
B: Why?
A: The trashcan is overflowing

Overfull means that too much has been placed in one location.
"Would you like some more roast beef?"
"No thanks - I am feeling rather overfull. I've eaten too much."
When things are 'piled up' they are layered on top of one another.
"The plates are piled up and waiting to be washed."

The first example is a straightforward way of saying this.
The second example is a more 'formal' way of saying this/

exceeded= gone beyond the limit.
Capacity= The amount allowed in a space.

"The trash can is more than full."
This means that the trash can very full and some of the trash is falling to the ground.

Your example can be used for this situation, although, I would prefer to say "the rubbish bin", there is nothing wrong in using the word "trash".