You can crack any case by elimination with evidence found at the location where the suspect died, their motives, which map to their modus operandi (e.g. how was the victim stabbed?), and their abilities (e.g. do they know the area well enough to hide from the cameras?). The algorithm is designed such that there is only one suspect who fits into all this, and this is the culprit.
This is a guide about one of the possible ways to solve a given case.
STEP 1: Identify the location of death.
Based on the other evidence found on the body, find and investigate the right location. You might find evidence relevant to the culprit's identity, e.g. perfume or gunpowder. Knowing whether the body has been moved can also help you narrow down your list of suspects.
STEP 2: Interrogate the suspects.
Start with those who are willing to talk. Their claims can make others' interrogations progress.
In L. C. Noire, suspects may lie or simply do not have the full information. If needed, search the home of the person they suspect to confirm the information they said. When you've gathered enough claims, guess the motive of the suspect by asking them about their relationship with the culprit (as seen in the image below). This will allow you to discover the true relationship and motive.
There is no penalty for a wrong guess for the question below. So, you can also try every option until you get it right.
STEP 3: Deduce the modus operandi based on motives and abilities.
The Glossary explains how motives and the modus operandi are usually related. For example, for someone with a lucre motive, you can exclude them if the victim's body has postmortem stab wounds or been tied up and stabbed, because these indicate strong emotions while lucre-driven modus operandi is more practical.
Some suspects don't have the abilities or the opportunities to use some weapons. For example, some poisons are inaccessible to some suspects. Suspects without much medical or anatomy knowledge wouldn't be able to cleanly aim at vital points either. So, you can exclude all of them.
If you still have any questions, feel free to ask them here!