Better questions produce better answers. At present, this question is quite broad, so the answer is quite general.
The exact rules changed over the 450 year history of the Republic. For most of that period, one had to be a member of the Senate to stand as a Consul. To be a member of the Senate required extensive land holdings and to be of noble birth.
The rules did originally preclude consecutive Consulships, so that the Consuls would not just replace the recently ousted Kings. But during times of peril, when it seemed that only one person could save them, the Senate decided to relax the rules. Of course they didn't tighten them after the crisis was over.
So it wasn't democratic as we understand the term, in a modern Representative Democracy. While there was significant separation of powers, the key way of winning an election was through public spending - what would now be called pork barreling, or worse.
A few references