The Perseids are actually caused by specks of dust released by the Swift–Tuttle comet. They always appear to be streaking out from the constellation of Perseus in the northeast of the night sky.
To see the Perseids in Gran Canaria you need to be somewhere on the island where you get an uninterrupted view of the northern sky with no bright lights in front of you.
Basically, the resorts and resort beaches are no good at all because of all the bright lights and because the Gran Canaria highlands block out lots of the sky.
However, all is not lost as there is a great spot for viewing the Perseids close to Playa del Inglés, and plenty of others around Gran Canaria.
The Fataga Road Viewpoint
The GC 60 road goes from San Fernando behind Playa del Inglés up to Fataga and San Bartolomé. About 10 km inland (20 minutes drive from Playa del Inglés) you get to the Mirador de las Lleguas viewpoint.
It's an excellent spot to view the night sky as there aren't many bright lights to the north and you are high enough to see most of the sky.
Beach viewing spots
Your best bet in south Gran Canaria are the beaches just east of Arguineguín like Playa Triana and Llano de los Militares. They don't have many artificial lights and are pretty flat so you can see a good chunk of the sky.
Head to the top
Anywhere at the top of Gran Canaria has the advantage of an uninterrupted view of the night sky, although the bright lights of Las Palmas do interfere with visibility in the northeast. A cool spot is the highest point on the island at Pico de las Nieves. Sitting next to Roque Nublo and spotting shooting stars is also an excellent way to spend a warm August night (take layers as it can be cold even during the summer).
Tamadaba is probably the best place in Gran Canaria to watch the Perseids because it doesn't have any artificial lights or light pollution from nearby towns. The trouble is that Tamadaba is a long way from anywhere so you are probably going to have to spend the night if you go meteor spotting.
That's not really a problem as it's a stunning spot (but you do need permission to pitch a tent and can't light fires during summer).
The Perseids in 2016
The Perseids this year are expected to be particularly spectacular with up to 200 shooting stars per minute at peak times (the average is 60 per minute). The best time to see them is between the 9th and 15th of August with the peak on the night of the 12th. You'll need to stay up until 02.00 in the morning as that's when the moon sets.