Can I copy Ulamog, the Ceaseless Hunger's ability twice with Strionic Resonator and Rings of Brighthearth?
Yes, this will work as described. You can use those abilities together to exile a total of 60 cards in one combat.

The exact sequence of steps plays out like this:


You declare Ulamog as an attacker, and its ability triggers.
While that ability is still on the stack, you activate Strionic Resonator's ability targeting Ulamog's ability.
Activating Strionic Resonator's ability triggers Rings of Brighthearth's ability.
Rings of Brighthearth's ability resolves. You choose to pay the cost to copy Strionic Resonator's ability, and you don't change the target.
The copy of Strionic Resonator's ability resolves, creating a copy of Ulamog's ability.
The first copy of Ulamog's ability resolves, and the defending player exiles 20 cards from the top of their library.
The original Strionic Resonator ability resolves, creating another copy of Ulamog's ability.
The second copy of Ulamog's ability resolves, and the defending player exiles another 20 cards from the top of their library.
The original Ulamog's ability resolves, and the defending player exiles  another 20 cards from the top of their library.