what animals alive today are the last surviving members of their entire clade?

The Northern White Rhino. There are two left (both female); so functionally extinct. Poached for their horns.

Clade is a tough subject because it does not specify how deep in evolutionary history we are grouping animals. 
With that said, 3 mammal clades survived the kt extinction, placentals have taken over, Marsupials are barely holding out, and Monotremes are all but gone. There are only 5 monotreme species left [egg laying mammals]. In the  future when Australia collides with Asia, i expect a similar result to when South America [rich in marsupials] collided with north America. 
Possums are the only conquerors in the Marsupial clade left. With Australia we shall see. Canids [Dingo], which were introduced by man, are already being attributed to the extinction of many animals in Australia, just how carnivora [including canids] are thought to have caused the terror birds near extinction in South America. We shall see

Humans are the last of the hominini clade.

Tiger.
Red tuna.
Asian elephant.
porpoise.
Mountain gorilla.
Irrawaddy river dolphin.
Rhinos.
Cheetahs.
Swordfish.
Elephants.

Humans.               

As member Ray implies, if you want your question to make sense, you'd need to specify "how deep in evolutionary history" you're talking about.  There are clades only a few hundred thousand years old, and there are clades hundreds of millions, or even billions, of years old.

The red panda is the only living member of the genus Ailurus and the family Ailuridae.