This change is an easy one to make - if you have human-readable files rather than the gobledygook that comes with the 1.1 update. Open up a capital ship file in notepad or whatever text program you like to use, then scroll right down to the bottom of the file. This is what the last few lines of the entry for the TEC Battleship looks like:

CommandPoints
 StartValue 0.600000
 ValueIncreasePerLevel 0.200000
weaponCooldownDecreasePerc
 StartValue 0.000000
 ValueIncreasePerLevel 0.030000
weaponDamageIncreasePerc
 StartValue 0.000000
 ValueIncreasePerLevel 0.050000
HyperspaceChargingSoundID "HYPERSPACE_CHARGEUP"
HyperspaceTravelSoundID "HYPERSPACE_TRAVEL"
EngineSoundID "ENGINE_TECHCAPITALSHIP"
formationRank 0
minShadow 0.000000
maxShadow 1.000000

The part that you're looking for is "weaponDamageIncreasePerc", which is 9 lines from the bottom in this case. As you can see, there are two more entries beneath it. The first one, "StartValue", is the standard rate at which the weapons fire - so it should be zero. But the second entry, "ValueIncreasePerLevel", is what you want to mod. The number is written as a decimal, so 0.05 would mean a 5% increase in attack power per level, 0.5 would mean a 50% increase, and 5.0 would mean a 500% increase.

Alternatively, if you worked with the entry above this, "weaponCooldownDecreasePerc", you can change the weapon fire rate, rather than the attack power. Ships that fire more often may do more damage in the long run than ships that fire powerful weapons less frequently. Both values increase when a ship gains a level, so you might want to consider modding both values. Again, to modify it, go to the second line below "weaponCooldownDecreasePerc", which is "ValueIncreasePerLevel", and change it to another decimal of your choice. I hope this helps.