Kachkar, 46, has pleaded not guilty to charges of first-degree murder and dangerous driving in the death of Sgt. Ryan Russell.
Russell died in the line of duty on Jan. 12, 2011, when he was struck by a snowplow in Toronto. He was just 35 years old.
The 11-year police veteran left behind a wife and young son.
Ontario Superior Court Justice Ian MacDonnell told jurors at the outset of the trial that "this case is not a whodunit" and that there is no dispute that Kachkar was driving the snowplow.
What is at issue is Kachkar's state of mind when Russell was killed. The Crown alleges that he drove at the officer with the intent to seriously harm or kill him.
'Not in a good frame of mind'
On Tuesday, jurors heard testimony from Kathleen Male, a woman who was staying in a St. Catharines homeless shelter at the same time as Kachkar in December of 2010.
Male testified that she often had meals with Kachkar and remembers him studying the Bible and attending daily devotion services at that time.
At one point, Male said that Kachkar asked her: "If you do something bad, does that mean you have to turn away from God?"
Under cross-examination, Male admitted that Kachkar wasn't clear on what he was referring to when he made the comment.
Male also testified that she remembers the day that Kachkar left the shelter.
"He was not in a good frame of mind, he seemed sad and downtrodden," she said in court Tuesday.
Male said she gave him a hug that day and asked him to come back. But she never saw him again.
"He seemed so sad."
Kachkar left the shelter on Jan. 6, 2011, six days before Russell was killed.
Kachkar had looked for work
Jurors also heard testimony from Nadine Janzen, an employment counsellor at a St. Catharines, Ont., employment centre who first met Kachkar on Oct. 31, 2008.
Janzen said that when she met Kachkar, he had just been laid off from a delivery job.
She referred Kachkar to a job-finders club and he was subsequently accepted for retraining.
Janzen recalled Kachkar as being "well-dressed" and "clean-looking" when he came into the employment centre.
She testified that Kackhar was "always polite, a pleasurable client to work with."
Kachkar wanted to get an AZ licence for truck driving, and later wanted to become trained as a heavy-equipment operator.
Janzen said there was a long period when she did not see Kachkar, starting in about September 2009. She eventually closed his file because she assumed he had found work.
But she saw him again on Jan. 7, 2011, just three days before he was due to come in for an appointment.
Kachkar did not make it to the appointment.
Another worker at the employment centre, Perry Bartley, testified Tuesday that he saw Kachkar twice in January 2011.
On the second occasion, on Jan. 7, 2011, Bartley said it seemed Kachkar's stress level was higher, he seemed more anxious and his lack of confidence was noticeable.
The defence asked if Kachkar seemed a bit off.
Bartley responded: "Yes."
The trial continues on Wednesday.