Dr. Armstrong was a wealthy and successful doctor, but he had a problem with alcohol. He operated on a patient several years back while under the influence of alcohol, and something went terribly wrong during the surgery. The patient, Louisa Mary Clees died as a result of Armstrong's error. A nurse who was involved with the surgery knew that Armstrong had been drinking, but she choose not to say anything about it.
Armstrong was hired by Mr. Owen under the impression that he was to secretely take care of Mrs. Owen as she was in poor health. Armstrong brought his medical bag filled with syringes and different types of medicine. This seemed quite suspicious to the other guests who later thought he might be the murderer.
One of my favorite books, And Then There Were None, is a wonderful story on how ten murders are forced to face the crime they had committed in the past. Trying to justify what they have done and be presented in good light in front of the society, each character in the novel tries to make out their situation in a way that makes them not see at fault. They don't want their status to be affected, so of course they are not going to admit to such crimes.
The crime Dr. Armstrong committed, as mentioned in the above answer, was murdering someone during an operation. Although accidents in surgery and in medicine occasionally do happen, Dr. Armstrong was drunk for this specific incident. That's a huge no no in the medical field. If he hadn't been drunk during the operation, perhaps the death could've been blamed on complications in the procedure, or how the patient reacted to certain movements and medicine. However, when you add alcohol into the equation, everything changes. Not only in this novel, but in real life too. For instance, a drunk driving accident is much more severe and illegal than your standard car accident. What Dr. Armstrong did, and reluctantly admitted, was an act of crime that society should not ignore. That is why Owen invited him to the island in the first place. To force him, and the rest of the group, to face the their committed crimes head on.
What led to the investigation you inquired about, as mentioned by the above answer, Dr. Armstrong was brought in by Mr. Owens to take care of Mrs. Owens. This was certainly seen as a suspicion because of the tools, medicine, and odd assortment of items Dr. Armstrong possessed with him on the island. Many of the other characters believed he was the murderer for this exact case. I hope this helps you with what you asked in regards to the investigation.