The short answer: patiently, and with lots of tinkering.
The long answer goes something like this: It is entirely possible to win games by conquest or domination in a Republic, but you have to be able to manage your unit costs and WW.
Unit Support -- You can help with unit costs by getting towns up to size 7 or above. Unit support in Rep is 1/3/4, so every time a town goes from size 6 to size 7, it triples the number of units it can support. Markets are also a big help. Once you get to 3 luxes, markets begin increasing the effectiveness of luxes. That may allow you to drop the lux slider a little. Finally, many players use what I think of as a "crust defense" in Republic. IOW, only border towns that the enemy can reach get defenders. If the enemy breaks through that crust, it can be painful, but units sitting in towns far from the front lines do nothing but eat up unit support and gold in a republic. I'll often place just a couple of defenders in the border towns, then back them up with a "zone defense" of fast movers that are capable of covering several towns at once. Once rails come in and the strategic railnet is laid, obviously, a stack can cover many more towns. Also, roads, roads, roads. Having lots of roads makes sense in any government, but putting roads on everything in sight is particularly important in a republic because of the commerce bonus. Once you put roads on everything, all of those 0 commerce tiles have 2 commerce!
WW -- I see that you've already discovered the joys of rock-throwing. Cats, trebs, etc. go a long way to managing losses and, consequently, WW. Roads will help generate commerce that can be spent using the lux slider. Having plenty of luxes and markets will also help. Keep wars short and to the point.
When I'm warring in Republic, I usually have a goal in mind, like a luxury or strategic resource, or even to push someone past a chokepoint (or off my continent). Once I've done that, I get ready to make peace. Now, sometimes the war goes so well that I'll press on, but once I've met that goal, if I need to make peace, I make peace. Also, WW is calculated separately for each civ, so if WW gets high against Civ A, you can make peace with them and go beat up Civ B for a while. So you can use oscillating wars to expand the empire. Go beat up Civ A & grab some land. Then go beat up Civ B & grab some more land. Rinse and repeat as necessary.