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.

• You could also try the slightly more micro thing of intentionally keeping a couple of other republics around. You can't build too many trade posts on your own, even if you make a giant dynasty. You can't steal from other families in your republic, if you have a much bigger trade empire than they have. But there is no limit on the cease trade post cb on other republics. I have controlled every sea zone in the Mediterranean with this trick. At some point the trade posts along make so much money the rest your vassals becomes irrelevant. The only problem was that my vassals kept annexing the other republics and that would destroy the trade posts. So I had to kill a lot of my own vassals to remove their claimants.
• You can kill every member of another trading house, that will redistribute their trade posts to the remaining houses. The doge at the time gets all their other holdings and money. This is really boring and slow though. Most trading families need two or three generations to really get somewhere. The other two stratagems allow you to gather power faster. It is also more difficult to pull off in the beginning, because the other houses are more equal in power to yours at the start.

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.