I’d start with dolphins, as they are already more than halfway there. Second would be dogs, of course. Because they are loving and loyal and would still care for their humans as a part of their pack, even if we were a little slow by their mental standards.
The sloth. Smart, but just as likely to take a nap rather than try to rebel against my rule.
As if they don’t feel superior enough already. You want to make them even more condescending? 
Rats.
Smart, quick to breed/mature, & has hand-like paws. Hands help with brains.
Or…just TOMATOES!
Fer sure! All they need, in addition to that, would be opposable thumbs, and they’re ready to conquer the world.
Oh wait! They’ve already got that! The end is in sight for humanity!
First thing I thought was hippos. They could easily create more chaos than any other animal.
Problem? When my enemies, the League of Goodness, called the army, my precious hippomen could be easily rounded up. They don’t reproduce easily. And of course I’m not going to clone them, because, hey, what sort of madman would do that?
So now I’m thinking cockroaches. Small. Strong. A couple of survivors from any attack and in a few days we are ready to go again.
Deer. They cause hundreds of fatal car crashes and hundreds of thousands of non-fatal crashes every year. Maybe we don’t breed them to be similar to a smart human, but at least smart enough to avoid on-coming cars.
Mad scientist or evil scientist?
Because I can think of little that would be more evil than to bring a sessile creature up to full sentience, like a barnacle or a tunicate.
There would be two things I would want in such a species: vocal apparatus that, in principle, could support speech; hands of some sort. Those two requirements would not seem to found together in any other species than H. sap. However, the apes have hands and could presumably learn sign, so that’s what I would go with.
One poster upthread mentioned Cordwainer Smith (Linebarger was his real name). But C’mel and D’Joan, not to mention E’telikeli could talk and had hands. That may be a much more difficult thing than intelligence.
Bird feet, particularly parrots with not just one but two opposable digits, can function pretty well as hands. Certainly, my parrots hold objects, manipulate them, and so forth with their feet.
I was leaning towards white mice until I read this.
Count me as being on the Future Planet of the Bears train.
I agree with Monty…ALL OF 'EM! I’ll be Doctor Moreau.
Come into my House of Pain, pussy-cat…
Parrots. They can already vocalize in a way that we understand. If they could form their own thoughts instead of “parroting” it would be awesome. Plus, to be super intelligent, and be able to fly is a bonus.
Giving the octopus human level intelligence (unless it’s Mississippi) is a very bad idea. The only thing keeping the murderous bastards from ruling the world now is that they’re squishy.
I can’t think of any animal that doesn’t have reason to hold a grudge against mankind. Dogs maybe but they’re so sycophantic that actually hearing them speak would make me uncomfortable.
Polar bears. Sure they may hate us but they’re close enough to extinction that we could take them in a fight.
Plus they’re cute and fluffy.
Monitor Lizards.
That’s my thinking. Give lizards sentience, give them a shot at running our government, and let’s see if they really can keep up a conspiracy. It’s a fool-proof plan, that only fanatical internet crazies will be able to see through.
The real answer is obviously chimps, and there most of the work is going to be on vocal apparatus, not intelligence, given the ‘verbal communication’ condition in the OP. I think a speaking chimp would already be able to convey self-awareness, today.
Having said that, I’m going with naked mole rats. Because I’m mad, innit?
#wrong thread
The tarsier would be a great one, birds and mice could be very useful for surveilance jobs. Dolphins and whales would make great team members with humans.
Humans.
Of course, I don’t actually hold out much hope for success.