Victor Klemperer's diary/book I Will Bear Witness answers this question for those one step before the concentration camp. The gestapo and Nazi government encouraged suicide in some discrete and cunning ways.

Long story short, this scholar was a good writer and kept a diary throughout his life. He had every 'special case' status in his favor, that let him be one of a few hundred German Jews who survived the war. He converted to Christianity in 1912(?), had a good WW1 military record, was not an immigrant, married an Aryan, was very lucky, and his university seems to have been courageous enough to let him keep a pension after he was no longer allowed to teach.

During the war, he lived in several "Jews' Houses" - basically public housing for mostly mixed couples. Full Jews without an Aryan spouse were mostly all sent to the camps much earlier.

At irregular intervals, the gestapo would come in for "inspections". Mostly, they were humiliated, robbed, beaten up, and subjected to all sorts of creative mistreatment so that when they finally did die or go to prison, it would look like their own fault. Just to illustrate, he nicknamed 2 regular visitors "the boxer" and "the spitter."

Anyway, a few policies seem to be pro-suicide. Sleeping pills in large quantites were always available without rations. When Jews could not buy meat, shaving powder, much soap, newspaper, toilet paper (making the no-newspaper thing even worse), etc., the sleeping pills were still a-ok. One specific brand got nicknamed "Jewish drops" because so many had overdosed.

He mentions prison cells for minor offenses would have a hangman's noose in it already, just in case you wanted to avail yourself.

In another entry, he mentions an elderly lady, 70-something IIRC, who was shouted at and beaten during a 'search' of her room. Afterwards, they produced a rope and showed her how to tie a noose. Afterward, they insisted she keep trying to tie one herself until she got it right.

It's impossible to tell definitely how many suicides there were for a number of different reasons.

In concentration camps proper (as opposed to death camps), especially before the war, when there was still a pretence to due judicial process, "suicide" was often given as the cause of death to the next of kin of concentration camp inmates killed by the guards.

In death camps proper no statistics were kept on the cause of death. This was because the cause was the same for everybody (gas) except for the tiny number of worker slaves kept around to dispose of the bodies and do other camp chores, who were ultimately destined for the gas chamber anyway. These camps were specifically built to kill Jews, no one was supposed to survive them and officially they did not even exist. The people were "disappeared" into these camps and never heard from again. No next of kin were notified, as the next of kin were the next intended victims themselves, insofar as they hadn't been killed already.

See for the difference between concentration camps and death camps this comment I wrote a while ago.

So all we have to go on are memoirs by survivors.

Concentration camp survivor memoirs mention most frequently suicide by walking into the electrified fences. Or just walking towards them, as approaching the fence was enough to get you shot. There is a tendency among survivors, especially those interned in camps for political reasons, to downplay the frequency of such suicides, perhaps to uphold the image of the heroic resister refusing to give up hope. And in reality, there were many other ways to lose your life if you wanted to in such a brutal environment where the slightest hint at insubordination could prove fatal. Or you could just stop caring, stop eating and stop paying attention to what the guards told you, which was the dreaded “Musulman” stage from which hardly anyone recovered. Why the inmates picked this word is not known, but it was used to refer to inmates who had become emaciated from malnutrition; ground down from overwork and had developed an indifference to life and a “thousand-yard stare”. They usually died very quickly afterwards, either because they could not keep up and were selected for the gas chambers or other means of disposing of sick inmates (some camps used lethal injections) or beaten to death on the spot by an irate guard, or through sheer exhaustion.

Death camp survivor memoirs (there are death camp survivors mainly because there were revolts at two of these camps that led to mass escapes) speak of very frequent suicides among the worker slaves in the early months of the camps' operation. In those early days the worker slave contingents were treated extremely brutally and regularly killed off and replaced by new workers from incoming deportation trains. That is until the Germans realised that they could run the camps more efficiently if they had experienced workers around that were fed adequately and treated a little better. But before that insight dawned on them, this is what happened in the worker barracks at night:

Those who could not condition themselves to this way of life--well, they left us during the nights. They committed suicide. They obtained sharp pieces of metal and slit their wrists. Their blood would drip down slowly on those sleeping on the bunk immediately beneath them. (Kalmen Wewryk - To Sobibor and Back: an Eyewitness Account)

Many suicides happened especially during the first night after their arrival.

We were taken to a special barracks where I was sleeping right next to the way. And there, at that night, it was the most horrible night for all the people. Because the memory, what people went through with each other, all the joys and the happiness, and the births and the weddings and other things, all of a sudden, in one second, to cut that through without anything, and without any guilt of the people. Because the people were not guilty at all, the only guilt for them was because they were Jewish people. Most of us we were all up at night, trying to talk to each other, which was not allowed. The commandant was sleeping in the same barrack. We were not allowed to talk to each other, or to express our view or our mind to each other. Until the morning, at five o'clock, we start going out from the barracks. In the morning when they held the appeal [roll call], to go out from the barracks, from our group, I would say at least four or five were dead. I don't know how the thing happened, they must have with them some kind of zyankali [potassium cyanide] or some kind of poison with which they poisoned themselves. Some of them were my close friends. They didn't say anything, we didn't even know that they had with them poison. (Abraham Bomba about Treblinka, in Claude Lanzmann's Shoah documentary)