Taking the capital is a great way to reduce cost of bribing other civ's cities. Remember that once a city is taken, whether by bribe or force, any units formerly supported by that city are lost.
So if you are able to save up a decent sized invasion/bribe force focused only on city capture, so as to nearly or completly conquer their cities in one turn, you won't have to contend with mopping up enemy units away from home----EVEN the ones in fortifications, or at sea.
In addition to taking the capital if possible or practical, a first turn strike should also include at minimum those other cities which support the most units, trade, wonders etc. so they can't retaliate effectively.
A quickly followed second or third wave are needed to complete the job, and make up for any mis-haps.
Your style will need dramatic changes and flexibility in MP games compared to SP games where the AI civ strategy is much more predictable.
If you can not do this in one or two turns, you may be in for a very costly campaign even if you enevitably win.
