To grow your own power, you need to upgrade your family estate and get your hands on some cities/counties. These will stay with your heir even if you lose power over the republic. So at the start place trade posts in counties where you want to use the seize city cb, something weak enough to attack, with a coastal city in it.
With enough expansion and upgrading of your holdings you should be able to get enough income to keep control over the republic by spending ~1000 on the election pool all the time. Since you can use the honorary title to designate any male dynasty member, if they are in your court, or your vassal as heir, a breeding program is quite easy to pull off. So you can always have a good heir.
From there if you are Christian you can use a combination of two things to become unbeatable. * Either you find a pope who likes you, then ask him for an invasion of a nearby kingdom. With that CB you get all the holdings in a county if you occupy the top holding. Use mercs and your retinue to occupy every holding in the kingdom, and there you will have it. A realm of same culture, same religion grand mayors to fill your coffers. Hire all the mercenaries and win the game.
In all honesty the trouble with playing a republic isn't that great. You cannot lose as long as your republic exists and you manage to produce at least one male heir every generation. That is all you need. So accept losing the elections some times. Try to make fun stories. Give away land when you are the doge. Change to a strange religion. Just try stuff, that is more fun than painting the map pink, or any other color for that matter.
The best games I've had as a republic were. One as the Catholic republic of Amalfi, while staying a loyal vassal of the orthodox Byzantine emperor. The other was as an awesome ass kicking viking republic Falkyrie, who raided everyone for money and had a harem full of foreign princesses.