In Heretics of Dune (1984), the Tleilaxu leader Waff thinks to himself:
No other people have mastered the genetic language as well as have the Bene Tleilax, he reassured himself. We are right to call it "the language of God" for God Himself has given us this great power.
and later, speaking to his compatriots:
"Is there one among you," Waff asked, "who believes that a single powindah knows how we use the language of God?"
A powindah is an unclean outsider. Waff is asserting that "how we use" the language of God is unique, not that the language itself is a Tleilaxu secret. Others, including the Bene Gesserit, know about genetics in their own way; the Bene Tleilax are the ones who use it to make gholas and Face Dancers and so on. In Heretics, a great deal of the plot is about the axolotl (or axlotl) tanks, held to be "their strongest weapon" which "they alone control".
As usual, this is made less mysterious in the spinoffs. Dune: House Atreides (2000) has directly:
Realizing the complexity of his challenge, the Master Researcher had brought in technological adepts from Tleilax One - learned men who had widely divergent views on how that goal might be attained. At this point in the process, all options must be considered, all evidence studied for clues to be inserted directly into the DNA code of organic molecules, which the Tleilaxu called the Language of God.
and in the comic version (issue 9, 2021):
We have brought in our best experts. They are studying clues in the deepest DNA code of the organic molecules... we call it the Language of God.
As to the actual technology or practice used, Chapterhouse: Dune (1985) tells us this:
Clever, clever, the Bene Tleilax. Far more clever than we suspected. And they have dirtied us with their axolotl tanks. The very word 'tank' - another of their deceptions. We pictured containers of warmed amniotic fluid, each 'tank' the focus of complex machinery to duplicate (in a subtle, discreet and controllable way) the workings of the womb. The 'tank' is there all right! But look at what it contains.
The Tleilaxu solution was direct: use the original. Nature had already worked it out over the eons. All the Bene Tleilax needed to do was add their own control system, their own way of replicating information stored in the cell.
The "Language of God," Scytale called it. Language of Shaitan was more appropriate.
Feedback. The cell directed its own womb. That was more or less what a fertilized ovum did anyway. The Tleilaxu merely refined it.
So their advantage is not the language of God itself, but the means that they have developed for genetic engineering, without needing a millennia-long breeding program to bring out desired human traits. Idaho has a memory in Heretics of being born from a tank, in fact a woman "monstrous in her almost immobile grossness... a maze of dark tubes linked her body to giant metal containers". It is not very precisely described, but the imagery supports a general idea of industrial technology imposed on the body.
This is a thematic contrast to the approach of the Bene Gesserit, and is meant to appear abhorrent because of the way it subordinates nature to technology and reduces a woman to an incubator. The Ixians are suspect for the same reason, making machines in the likeness of a human mind (or almost). While the gholas in particular display unnatural abilities, in terms of their memories, this is not something that the Tleilaxu have invented, but something they have accessed via a "shortcut" (the word used in Heretics). The books do not describe the processes in detail, but the thrust of the presentation is that whatever it is, is industrial. They surround it with religious reverence but at its heart, the practice is a mechanical one.
