When someone accuses you of wrong doing after you had accused them of something, what do you call that or them?
This is a Recrimination:
an angry statement in which you accuse or criticize a person who has accused or criticized you
As Tonepoet mentions in the comments below, one who recriminates is a recriminator. While the definition of recriminator perfectly fits the person you describe, the word's rarity makes me hesitate to recommend making use of it (recrimination is more common).
It's quite common to see this called exchanging accusations, trading accusations or tit-for-tat accusations (or simply "tit-for-tat"). The word "accusations" can be replaced by "allegations" in any of the previous phrases.
The treasurer’s race turned into a harsh and confrontational battle late in the campaign, with the candidates exchanging allegations of ethical misconduct and claims that their opponent was distorting the truth. — Gregory B. Hladky and Jon Lender, "Nappier Declares Victory In Tight Treasurer's Race; Herbst Concedes", Hartford Courant (5 November 2014)
Allegations that French far-right right MEPs have breached European Parliament rules on hiring assistants have triggered “tit-for-tat” accusations among other MEPs, the director of the EU's anti-fraud agency told journalists on Tuesday (2 June). Giovanni Kessler likened it to the wave of accusations among German politicians that followed a 2011 plagiarism scandal involving the then defence minister Karl-Theodor zu Guttenberg. “They started accusing each other ... This is what is happening in the parliament now”, said Kessler, who is director general of the European Anti-Fraud Office (Olaf). — Peter Teffer, "MEPs trade fraud allegations over assistants", EU Observer (2 June 2015)
A phase for the process of exchanging accusations about who is at fault for a specific event is blame game, whereas in the previous phrases the two accusations may not be linked.
blame game: accusations exchanged among people who refuse to accept sole responsibility for some undesirable event — The Free Dictionary
A good word for the allegation that has been made in response is a counter-allegation (often written without the hyphen):
Prosecutors may often be presented with conflicting accounts of the incident, with each party claiming to be the victim. The defendant may make a counter-allegation of abuse or violence, or argue that s/he acted in self defence, making it difficult to identify and distinguish between the victim and offender. — "The prosecution of domestic violence cases", Crown Prosecution Service (2014)
There is also counter-accusation:
The latest events have brought accusation and counter-accusation. Kyiv blamed pro-Russian protesters for attacking pro-Ukrainians. “The provocations that happened in Odessa causing clashes and many victims were the result of outside interference. They were financed by former members of the government of (deposed president Viktor) Yanukovych,” said Kateryna Kosareva,spokeswoman for the Ukrainian Security Service. A spokesman for the Kremlin blamed Kyiv and what he called its “Western sponsors”. — "Ukraine and Russia blame each other for deadly violence in Odessa ", Euronews (3 May 2014)
In more colloquial terms you might say that you are passing the buck, laying the blame, or deflecting criticism.
While these phrases CAN be applied to the described situation, they don't inherently imply that you are accusing the person who directly accused you (A blames B, so B blames A), and can be used in situations where A blames B, so B blames C. They also typically imply that that A's accusation against B is invalid or less severe due to whatever B is accusing C of.
This is sometimes called the tu quoque fallacy.
Aurast's answer describes the statement, but you might describe the person as vindictive:
having or showing a strong or unreasoning desire for revenge.
"the criticism was both vindictive and personalized"
synonyms: vengeful, revengeful, unforgiving, resentful, acrimonious, bitter; spiteful, mean, rancorous, venomous, malicious, malevolent, nasty, mean-spirited, cruel, unkind