There's no one performance metric you should focus on. What you should do is to formulate a game plan based on your civ's strengths, and target metrics that are appropriate for your situation. I'll share some examples.
If you're planning to win a culture victory as Russia, your strategy may involve generating significant tourism from early on in the game by hoarding great works. You can generate relevant GPPs through Lavra and theatre square buildings and place the works in the latter. Fast expansion is crucial to this strategy because the number of such districts you can build is limited by the number of cities you have. Russia also has the problem that its cities don't tend to grow quickly, so in many instances, you'll have to choose between the two districts, making quick expansion that much more important for Russia. So, what would be a good performance metric for Russia? It changes throughout the game. At the beginning, you want to make sure your first 2-3 cities are capable of churning out settlers as quickly as possible. What do you need for that? Faith and production. Your first cities should be in tundra and prioritize building Lavras and shrine. Afterwards, as you start to recruit great writers, you'll want to make sure you have the slots for them, so the number of great work slots could be an important metric.
Byzantium led by Basil, in my opinion, is the most straightforward domination civ, but not because it follows the generic "10 cities by 100 turns" advice or anything of that sort. Early-game Basil only cares about culture. You don't need much science because Tagmata are unlocked by a civic. You don't need gold because Hippodromes give you free units. You don't strictly need a great general (although it's nice to have one) because you get significant combat bonus from religion. You don't even need faith to spread your religion to take advantage of the Crusade belief and Taxis. You could try to maximize culture per turn by settling cities because that allows you to build more culture-yielding infrastructure like monuments or possibly shrines if you have the Choral Music belief. Or you could try to spread your religion quickly and leverage the World Church belief. One way to do this is to get a couple horsemen early and pick on a neighbouring civ or city state and focus on killing units. You could also try to pillage for culture and faith. Use the extra faith to get a quick apostle to grab the Crusade belief to make your horsemen near invincible. If you're going down this route, you may find it useful to try to get an early general. Thinking about how many cities you should have by what turn doesn't really help you succeed with Basil. In fact, it should be possible for you to generate enough culture operating with 3-4 cities, unlock Tagmata reasonably quickly (~90 turns?) and get to a stage where you can wipe out an entire civ in fewer than 10 turns.
Korea's Sejong is unique in that he wants to track the metric of maximum science per turn per culture per turn (or just the ratio between science and culture per turn). This civ's strength is largely determined by how many turns you can pull forward key civics, and that's based on the science/culture ratio. Pre-Feudalism, you want to have a small number of cities (3, maybe 4), prioritize Seowons and libraries, while not neglecting culture too much without going out of your way for extra culture. This means just utilizing the usual monument + Pingala combination. In my experience, aiming for a ratio of 3-4 works well for the Classical and Medieval eras, allowing you to pull forward 12-16 turns worth of culture in total. You really don't want to be settling too many cities early on with Sejong because you won't be able to focus on maximizing your science output and getting all the necessary inspirations before you absolutely need to clear a new era tech. That restriction is lifted once you enter the Medieval era because there will be more techs to clear, and with the Ancestral Hall-Serfdom combination, you can more efficiently settle more cities once you're in the Medieval era. This might mean you're operating with only 3 cities until around turn 80, which again, goes against the popular advice, but I think it's hard to argue you're far behind when you've cleared both Apprenticeship and Feudalism by that point of the game.
I think the best way to improve your game is to take a civ that has a straightforward early-game path and perfect a build for it. I introduced three here (although I don't recommend Sejong because he requires a ton of micromanagement), but other examples include Ethiopia, Gran Colombia and Japan's Tokugawa. Think about how to maximize these civs' strengths as quickly as you can.
and then eliminate them and become a much better player no of cities by turn 100 would be far easier to measure 
