The reason that spokes get broken is typically low tension, not high tension. If your spokes are at too low a tension right now, then raising that will make them last longer.
OK, so we've got two likely low tension scenarios here that are causing your problem:
1. Uneven spoke tension. If you pluck each of the rear spokes on the DS of your rear wheel and one or two sound really high pitched and one or two sound dead/low pitched, then you've got uneven spoke tension. The low pitched spokes will be under more stress with every wheel revolution and will break much faster. You want to fix this. So, to even out tension:
- Go all the way around the DS and all the way around the NDS and find the deadest/lowest pitched spokes. Get some masking tape and mark these spokes.
- Check the spoke directly in front of and behind the dead spoke on the same side. If they're a higher pitch, then back both of them off 1/8th of a turn, and tighten the dead spoke 1/4 turn.
- Go around the wheel a couple times, working on spokes of one side and then the other until things are starting to sound more even all the way around. At this point your wheel will have started to go out of true.
- If your rim is in OK shape, you should be able to get the wheel back into true by tightening groups of spokes in groups of three or more on a side . . . so if you find a little bump in on the NDS side, then tighten the spoke closes to the bump 1/4 turn, the spoke in front of it 1/8th, and the spoke behind it 1/8th. You can use smaller amounts of tightening as you get closer to true, and if you're a perfectionist, this will start to take a long time.
2. Overall spoke tension is too loose. If your rear wheel is reasonably true, the DS spokes all sound roughly the same pitch, and the NDS spokes all sound roughly the same pitch, then try increasing tension on the whole wheel:
- Start on the DS at the valve hole (so you remember where you started). Tighten each DS spoke 1/4 turn.
- Go to the NDS at the valve hole. Tighten each NDS spoke 1/4 turn.
- Check for true of the wheel and if it's starting to go out of whack, fix it (as above, you want to fix this by adjusting groups of spokes, not individual ones to try to keep tension more even).
- Keep doing this until your NDS spokes feel tight. You want to be able to grab two of them, squeeze 'em between your hands and feel some decent resistance.
You're not going to get it perfect without the proper tools, but this should hopefully put you on the right path. After you finish truing the wheel, take the bike for a short ride around the block and then check the wheels for true again (they often go a little wonky after they're put under tension/stress following a truing) and fix any bumps you find.
If I were you, I'd buy a bunch of spokes the correct size for your DS and NDS of the wheel and keep 'em handy so you can swap them out on their own. Spokes are usually only a dollar or so a piece, and once you get good at it replacing a spoke and bringing the new one up to tension only takes 10-15 minutes.