I believe you can just modify CIV5GameSpeeds.xml to suit your tastes. It will be in the Assets\Gameplay\XML\GameInfo folder of your Civ 5 folder.
Each difficulty level is listed, with a bunch of percentages that are applied to the cost of everything. The ones you are likely interested in are TrainPercent (presumably affecting the cost of units) and ConstructPercent (presumably affecting the cost of buildings). I'm not sure about CreatePercent, I think that may affect the cost of Wonders, but I'm not certain. And there is also BuildPercent, not sure what that is. So just take a text editor, adjust the values for the difficulty level you want to play, and then start a new game.
I guess the other option would be to start with the normal game length and just increase the relatively obvious settings for Research and Culture percent (if that really is all you want to slow down). I have a feeling this won't necessarily give the feeling you want though, you'll just end up with a game that feels like it is moving fast but you can't research anything.
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.
Best Answer
If I'm going for a conquest victory, I tend to try and focus on researching the techs for siege weapons early. You can attempt to siege a city with just archers and melee units, but I think the city defensive bonuses make this a tough fight.
While you're working towards catapults, create some early melee units. Send them out to gain experience, and then upgrade them when possible, you'll have the backbone for a strong early game army.
In the very early going, I'd focus more on grabbing land and stabilizing your economy, so that when you make the shift to military production, you can still support your existing cities.
On the culture front, take the "Honor" track, and grab a Great General early. If you keep the General with your siege units, and properly protect them, you'll also have a significant advantage.
Being at war will mean you're sometimes on the defensive - make siege units for your border cities with your current enemies. One or two siege units can do devastating damage to AI armies, who don't tend to assault with enough strategy to avoid your siege weaponry. Sometimes it's even better to declare war, and then sit and wait for their armies - you can gain free experience and decimate their fighting forces with minimal effort.
Specializing cities also helps; this was something I learned late in my Civ4 days. Try to found your second or third city someplace with high production value tiles, such as hills and forest. Focus this city on unit experience and production speed bonus buildings. This will keep your war machine in good shape even into the mid and late game.