Speaking from the perspective of the majority of the population...
This sentence is actually the better option, because it correctly links the noun phrases, and it is not recommended to change it.
Speaking from the majority of the population's perspective...
This sentence is actually not so good, because the "'s" is here attached to "population", which isn't really what you want. Consider the following:
Speaking from the advocates of corporal punishment's perspective... [Very weird if not wrong!]
In general you don't need to always avoid repetition of "of", unless it becomes too many, in which case you can avoid it by changing the focus, such as:
For the majority of the population, their perspective is that ...
Or you can say the same thing in a different way:
To the majority of the population, [however,] ...
Here are some perfectly acceptable examples of repeated "of" from BBC:
deadly grip of birds of prey
the story of one of the literary treasures of the medieval world
the contributions of some of the giants of mathematics
over the course of hundreds of thousands of years
the sequence of bases of a section of DNA