The question in the headline is kind of a loaded one.
Bizarro is, by his nature, a character somewhat unlikely to show up in the kind of superhero movie Zack Snyder likes to make. In most iterations, he's only really a threat because he fails to understand or properly interact with the world around him -- and he engenders some sort of pity on the part of even those who are tasked with bringing him down.
That said, who or what, exactly, is Bizarro?
Let's start at the beginning.
Bizarro debuted in 1958's Superboy #68, a Frankenstein's monster pastiche with all of Superboy's powers created by accident when a scientist was demonstrating a "duplicating ray." In the Silver Age, the creation of the first bizarros took place on Krypton; it was creating a mindless army of imperfect clones of himself that landed General Zod in the Phantom Zone.
On Earth, Bizarro is a flawed imitation of Superboy, with chalky white skin and childlike, erratic behavior. At the end of his first story, Bizarro died; while technically Superboy brought the instrument of Bizarro's destruction to the battlefield, the creature actually willingly sacrificed himself, the ashes of his body restoring sight to a blind girl Bizarro had briefly befriended in Smallville.
The character proved popular enough to bring back, and for most of the Silver and Bronze Ages, Bizarro was created by Lex Luthor. Turning on Lex, Bizarro helped Superman arrest the villain but later used the duplicating ray to create an entire world for himself, full of bizarro duplicates of Lois Lane, Jimmy Olsen and more. The "Bizarro World" version of the character was the longest-lasting and best-known until he was wiped out of existence and removed from the history books by the post-Crisis on Infinite Earths Superman reboot.
In The Man of Steel, John Byrne introduced a new version of Bizarro created by Luthor for the first time. In the post-Crisis continuity, Superman never had powers as a boy, so the '50s origin couldn't have happened. Lex created Bizarro not with a duplication ray, but as an imperfect clone of the Man of Steel. There, it was Lucy Lane who was blind, and whose eyesight was restored when Bizarro was destroyed. Variations on this Bizarro would appear twice more: In 1994's "Bizarro's World" storyline, during which Luthor once again created a Bizarro in order to research potential cures for a plague affecting Metropolis's clones; and again in Superman Forever, when Luthor's ex-wife used his files to create a Bizarro. In both cases, the creatures would be destroyed or otherwise die during the stories that introduced them.
Another Bizarro would later be created not by science but by the nigh-magical powers of Mr. Mxyzptlk, a fifth-dimensional imp, while they (the powers, that is) were being controlled by The Joker. This one would survive for a while, taking part in Infinite Crisis and Blackest Night, among other big stories. That version of the character would have the ability to create new Bizarros, leading to a new Bizarro World.
Then came the post-Flashpoint reboot. The current Bizarro was again a failed clone of Superman, this time using DNA from Superman and a human. Ironically, that's how the post-Crisis Superboy was successfully genetically-engineered. That version of Bizarro has incendiary breath and cryonic vision (rather than Superman's freeze-breath and heat vision), and feeds of Kryptonite, but is injured by solar radiation. That character is currently dead in the comics, having died in Forever Evil, but he'll be back soonish, as Lex is attempting to clone him again -- and this time wants to purposely recreate Bizarro, not clone Superman.
Introduced in the best-selling "The Death of Superman" storyline in 1992, Doomsday is a genetically-engineered killing machine born on ancient Krypton and rocketed into space millennia ago after he killed his way across a number of planets. His only real tie to Lex Luthor is that in Superman: Doomsday, a direct-to-DVD adaptation of the storyline, Luthor's employees are the ones who inadvertently discover and release Doomsday.
But in the trailer for Batman V Superman: Dawn of Justice, we seem to see Doomsday being created as a result of genetic tinkering where Lex is working with the body of General Zod.
So...Zod, whose genetic clones were the original bizarros, is being used as genetic fodder for Luthor's experiments in Batman V Superman, apparently under similar conditions (he wants either his own Superman or an anti-Superman insurance plan in place) that the first pre-Crisis on Infinite Earths Bizarro was created in The Man of Steel.
In Superman/Doomsday: Hunter/Prey, there's an interesting observation about Doomsday. It's there that it was first revealed that Doomsday was created on ancient Krypton, and in that story, it says that while Doomsday left the world he was born on, that planet's own sad fate would be linked to the genetic experiments that created him.
(I'm paraphrasing, so forgive me if that's a little bit off.)
That suggested to me at the time that the clone wars that destabilized Krypton's government and society, leading to a rift between the spiritual and scientific elements of Kryptonian culture as described during the Exile in Space storyline, had roots in the technology Bertron had left behind after Doomsday killed him and his fellow scientists. It's interesting, then, that an early Bizarro story established that the first "bizarros" were clones of Zod, created on Krypton while it was still alive.
In the film, of course, it seems as though most of that Bertron stuff wouldn't apply. Doomsday, after all, is seemingly created by Luthor, right?
Well, on the Blu-ray for Man of Steel, Bertron gets a name-drop in the special features: "Beware Bertron's curse, for he is named Doomsday." These are the same bonus features that referenced Wayne Enterprises onscreen for one of the first times in the then-fledgling DC Extended Universe.
Could Luthor be somehow using Bertron's data to successfully clone Zod? And if so, how much is this really Doomsday, and how much is it Bizarro?
It's been noted that the bony protrusions on Doomsday's body are suspiciously small in the image we saw of the monster in the Batman V Superman trailer. Many fans online have noted that the rapid evolution that is one of Doomsday's powers could easily give them a more "familiar-looking" Doomsday by the end of the film.
But what if the protrusions on Doomsday's face and joints turn out, in this case, to be a version of the crystalline look many versions of Bizarro take on as they decay? Could the secret to defeating Doomsday, then, be to simply wait until he becomes sufficiently "bony" to be pounded into dust like so many Bizarros before him?
In any event: Will we see Bizarro? My guess is "kind of." I can't imagine anyone will ever refer to anything created onscreen in this film as "Bizarro," although there could be earlier, pre-Doomsday clones that look something like Bizarro when they fail.
Meanwhile, a version of Bizarro absolutely will be appearing on CBS's Supergirl this season. So there's that.