I've played in and run evil campaigns of various sorts in both 3.5 and 4e (though not 5e, I think my learning will transfer), and run into a lot of problems: My Guy Syndrome comes up a lot, as does a tendency to default to a regular D&D storyline only with more stealing of spoons and kicking of puppies to remind ourselves we're evil. Sometimes an evil campaign instead descends into over-the-top motiveless violence until there's no story at all. There's a whole host of at-the-table and in-the-story issues, and I tried many different strategies to address them. Eventually I came up with a framing device which works well for us in avoiding these problems:
Provide the PCs with a Master to guide them toward orchestrated works of Evil.
Start the game with the PCs as underlings/minions/hirelings/apprentices/etc of a powerful evil NPC. The Master has a complicated Evil Plan and he tasks his minions to enact various parts as the Plan progresses: "Bring me the soul of a hound archon," "Raze the border keep," "Steal the Apocalypse Gem," "Help a spy infiltrate the paladin's ranks," and so forth, tailored to the PCs' abilities.
This provides the party a reason to work together despite having different agendas (and working together will hopefully bond them as friends so that they want to continue as a group) and establishes small achievable evil goals that accumulate into an Epic Evil Event.
All you need to do is ask the players to make sure their characters have a good reason to work for the Master: The serial killer likes having his rampages subsidised (and the Master protects him from the Law); the necromancer seeks to learn from the Master's experience and gain access to his libraries of forbidden lore; the mercenary's in it for the money and benefits.
Eventually the Apprentices will surpass their Master.
Expect the party to betray their Master at some point, hijacking his Evil Plot for their own gain: this is not only expected, but awesome. It's the Master's Evil Plot, not yours, and the story isn't about the Master--it's about his apprentices. Consider the Master to be training wheels for evil, setting an example which the party can then follow to surpass and overthrow their instructor as they level up.
This works because Evil Needs Goals.
As Ed describes so well and AgentPaper elaborates in the D&D context, evil needs concrete reasons motivating its actions. The Master provides goals and motives while the players find their feet in the new paradigm, channeling and guiding their exploration of what it means to be evil in ways compatible with the D&D paradigm without simply kicking puppies during a dungeoncrawl.
A word of warning: Alignment is tricky.
D&D has a history of the details and nature of alignment sparking major heartfelt arguments, because D&D alignments are not easily (or appropriately) matched to real-world philosophies and moralities; they're narrative simplifications to support the game's conceits and draw their power from storytelling conventions rather than from genuine moral complexity. Exactly what this means and how to deal with it are beyond the scope of this answer (and possibly this site, although there's a LOT of questions on the topic you can look at), but you should be aware it exists and be ready to talk with your players about what "Evil campaign" means to them so there aren't nasty surprises mid-game.
Handle Animal covers everything you need to know.
The Handle Animal skill allows the player to train animals to learn specific tricks and abilities as defined in the skill description, as well as handle them as pets. You can also teach general purpose commands if you want the animal to perform a specific role in the party.
You are trained at working with animals, and can teach them tricks, get them to follow your simple commands, or even domesticate them.
-Core Rulebook, Skill Descriptions, Handle Animal (emphasis mine)
Ultimate Equipment also covers the general list of pets a player can purchase as well as their respective prices. Statblocks are not provided, but instead must be located in the Bestiary entry of the desired creature.
Best Answer
Barring in mind that in Pathfinder, and D&D as a whole, Good and Evil are not just concepts, but measurable, detectable, fundamentally defined forces of the Universe; such acts will draw attention from extra-planar beings.
Setting aside what the player/character has said to other players/characters, ask the player what his character is truly thinking he is doing. Is it simply because they are evil, and that is how the world works? Is it because he just wants to kill, and evil creatures are justifiable targets in the grand scheme? Or does he actually feel that allowing them to exist will only cause destruction and mayhem, and any blood they spill will be on his hands, metaphysically speaking.
If the first, either explain to him that is not how it works and continuing to do so will have consequences, possibly both through alignment shift and in-game NPC interaction. One example would be shopkeepers and merchants refusing to do business with a cold-blooded killer or anyone they associate with, thus impacting the entire group. Maybe the local constabulary has a "reward" for the character that has to be claimed in person, that is actually a pair of manacles and a jail cell.
If the second, have a devil, or other Lawful Evil outsider, contact him when he is alone; either on watch or in dreams, est. Have them make promises of power and reward for not only continuing his actions, but convincing others to copy him. This provides the opportunity for role-playing an individual with real issues. If you are not comfortable with actually giving him anything, have it be hallucinations to further drive home the point that what he is doing is crazy.
If the third, have a Good aligned NPC or outsider make contact and explain that while they appreciate the intentions, and applaud the effort he is putting in, he may want to dial back the zealotry a tad, maybe include a story or proverb about a historical figure that underwent a similar campaign. Or, instead of requesting him change his intensity, have them ask to change his directed target. Gobins are bad, but nothing compared to the demon-summoning, undead creating, walking personification of Evil that is tyrannically ruling the kingdom to the [pick direction] and is gearing up for invasion of the party's current location.
In summation, talk to the player, and act accordingly to his response.