The tactics for winning as an underdog are not that different from the tactics for winning in general. Check out agent86's excellent general advice for beginners.
All advice should be put in perspective with the difficulty level. At 3 and below, you can sit on 3 to 5 cities and focus on food and happy to just outgrow the AI. At 4, the AI plays fairly - the main deficiency in the Prince AI is unit management. At 5 and higher, the AI's get extra everything and cheat! It can be very difficult catch up against King or better AI's.
Some civ's are early peakers. Germans, Russians, Mongolians will expand as much as they can at the start.
Fortunately, score only matters if and when time runs out. There are four other ways to win the game (military, cultural, science, diplomacy).
If you are truly behind, cultural victory is a great equalizer. You don't need to grow cities, just try to keep up in tech and focus on art. Cultural victory is so fast that the only two ways you can lose are: lose your capital or lose to another cultural victory.
If you're just behind on score, but still ahead on hammers (and player brains), you could go for military victory. You could have the lowest score, but if you have all the capitals, you win.
Diplomacy and Science victories are usually a way for the score leader to win (shortly) before time runs out, I wouldn't recommend going for those.
If you do decide to catch up through conquest, it may be better to absorb (or raze and replace) the cities of a weaker civ than to directly attack the leader.