I just won an influence victory. I thought i would share how i did it to everyone.
First of all the first rounds count bigtime. I Usually research Xeno labs first, then go right for advanced propulsion until i get ion drives. I constructed a custom colony ship (called Shock Colony) that travels 2x the speed of normal colony ships and carries the same number of people. While doing this research, i bought one factory on the very first turn. Every single turn until i ran out of initial credits i bought colony ships. Do whatever you have to do to expand as fast as possible, gaining as many systems and planets as possible, and of course, break for high numbered planets.
Its important not to ignore low numbered planets too. They are just as important as the high numbered ones in the long run.
Here is the real sneaky secret. Build Embassies on every planet, in large quantities. Not overwhelming amounts so it ruins your growth, but plenty, say, 5-6 on a large numbered planet, 3-4 on medium, 2-3 on small, 1 on really small. Build farms, around 3 or so on planets that can hold em.
And influence starbases, 1 for every star you have a planet orbiting. you don't need to place it right next to the enemy planet.
Research ALL influence techs, specifically Xeno business and Xeno Government lines.
ANY AND ALL Influence resources that you can find MUST be aquired through PEACEFUL means. and do NOT research weapons techs until they are down to 1 turn researches.
War can really screw up your plans, since your focusing so much on influence.
Of course, when your influential, convince neighbors to go to war with your enemy...
an important thing: Never. ever. ever. ever. EVER. give a diplomacy or influence tech away for ANY reason. economics, weapons, and planetary improvements are reccomended (Since it reduces the amounts of upgrading needed when you steal systems from them).
Do this and you will watch your colors sweep the map!
One last thing, if you want weapons techs, get it from minor races, and use any tech you want since influence doesn't do a thing for them