The grammar is Russian; the vocabulary has changed over time.
The original fenya consisted of broken Russian words, words borrowed from Greek and other foreign languages. Vladimir Dahl in his monumental Explanatory Dictionary of the Living Great Russian language gives the following examples of ofenya's parlance:
"Ропа кимать, полумеркот, рыхло закурещат ворыханы."
Normative Russian: "Пора спать, полночь; скоро запоют петухи."
Translation: "Time to go to bed,
"Да позагорбил басве слемзить: астона басвинска ухалила дряботницей.
Normative Russian: "Да позабыл тебе сказать: жена твоя померла весною."
Translation: "Oh, yes, I forgot to tell you: your wife died this spring"
The vocabulary changed over time, with notable infusion of words of Yiddish origin. During the times of the Soviet Union fenya penetrated into common spoken Russian and can no longer be considered cryptic, although it is still commonly associated with those who have connections to the Russian criminal culture or who have spent a significant amount of time incarcerated. A number of explanations for this phenomenon are suggested. For one, a significant part of the population, not necessarily criminals, went through labor camps, and massive indiscriminate amnesties after the death of Stalin resulted in a penetration of the subculture of convicts into everyday life in the form of a shock wave. Also, the criminal life was romanticized in popular culture: for example, in the form of "blatnaya song", see Shanson. Few "common" Russians possess a complete or even complex understanding of it and fewer still - for various reasons - will admit to it.
Fenya influences the Russian culture in different ways. In particular, a whole subgenre of Russian humour exists, in which a known tale, such as Romeo and Juliet or a popular Russian fairy tale is cast into fenya.
The collapse of the Soviet Union and the appearance of "New Russians" introduced a new changes into fenya, notably assigning new meanings and accents to common words.