The Death Note manga and anime can be seen as a moral study of a young man's decaying ethics following his acquisition of immense power. Starring a high school student who gains access to a literally killer book, the series eventually becomes a cat-and-mouse hunt for its protagonist serial killer.
Light Yagami is hunted down by the authorities for his developed taste for wanton and convenient killings, but he initially desired to kill only crooks. While Light's initial goals may have been noble, his descent into serial homicide is one that seems inevitable given the circumstances. Still, many fans might question: was there ever a scenario in which Light could have stuck more closely to his original goal? What might have happened if he hadn't caught the eye of the authorities?
Kira's Initial Killings
The series begins with Light Yagami finding the Death Note and questioning whether or not it's legit. However, after using it to cause the death of a would-be criminal, he believes in its power and wonders whether or not he should continue to use it and kill evildoers. He comes to the conclusion that no one else in the world is as well suited for such a task as him, and he perseveres on his newfound mission of cutting down evil weeds wherever they sprout.
Becoming known by the name of Kira, the vigilante Light racks up an immense body count in a relatively short amount of time. They're initially all criminals, like Light set out to get rid of, but eventually, the morality behind someone of the killings becomes increasingly questionable. It reaches a point where Light even uses the Death Note to kill people simply so as to evade being caught. This draws the attention of detective L and the Japanese Task Force, who investigate who's behind the wave of killings, and these would've continued without their intervention.
Kira Was Destined to Become Evil
This hunt for Kira and the final events of Light Yagami's life all stem from Light supposedly going too far and stepping outside of his initial parameters of only killing those who are most deserving of righteous judgment. In all honesty, his egotistical downfall is set in motion once he decides to use the Death Note, and the authorities' interference would not have changed this.
Light is an incredibly intelligent individual with a strong sense of justice, and these two traits arguably make him into someone with a black-and-white, self-righteous sense of morality. Combine this with the power that the Death Note gives him, and he would have eventually killed literally anyone who questioned him too much.
He also quickly comes to the conclusion that only he is suited for the role of judge, jury and executioner, and he soon takes this to an extreme when he proclaims that he will use the Death Note and its power to become the god of this world. His mentality was already warped prior to using the Death Note, as he quickly abuses its power for whatever means he sees fit, and he soon loses any sense of his old morality and grip on reality. Keep in mind that this descent into the mindset of a world-dominating killer came before any of the authorities began investigating Light. This egotism is not simply in response to being challenged by L and the Japanese Task Force, so said rivalry is essentially a stumbling block in his path of conquest.
In regards to his original goal, it's uncertain if he would have truly been able to kill off all evildoers, but an unchallenged Kira would have definitely been able to rack up an even greater body count. It's unknown what aims he would have gone to in order to sate his self-righteous bloodlust, but what it undoubtedly would have seen is an even greater amount of carnage. As shown once the Task Force is introduced, Kira is not above killing people simply to throw the authorities off of his scent. This shows how he could have easily taken it upon himself to just kill people over the slightest infraction, having little regard for true good and evil and merely removing those he deemed a nuisance.
His rivalry with L is also stoked by an appeal to said ego, with Light more angered by being used than he is over senselessly killing a decoy. This also shows that morality, ethics and right and wrong are out of the window at this point and that Light is now only interested in asserting his power and dominance. Plus, L's back and forth with Kira distracted the killer from wielding his power even more bluntly. This pride ultimately comes before his ultimate fall, with the would-be god of death succumbing to his own loss of life, yet without that interference, Light may have been able to focus more on his deadly plans.