Just for starters, I really don't think inflation could be integrated in its actual form. The idea I am suggested just popped into my head, but I probably would not support it.
Here is how to integrate a very superficial idea of inflation into civ. Remember, it doesn't really reflect what inflation is, just what most people think it is. A couple tenets first:
1) All civs have their own currency, which will have a relative value vs. Gold, the standard.
2) Unlike reality, positive actions will make your currency more valuable, negative actions will make your currency less valuable.
3) It is assumed nations can print or coin excess money whenever they need once Currency is discovered.
Imagine the world currency market(just for example points) at 4000 BC is Zululand, Rome, English, Novogorod.
Gold
Shaka = 100 for 1g
Drachme = 100 for 1g
Pound = 100 for 1g
Potato = 100 for 1g
Now here is where players could bend money, to an extent.
King George does not have enough moeny to pay for all the improvements he built, so he prints money. It works easily on the local level, but decreases the value of the Pound vs. Gold, which in turn makes his money less valuable to other nations.
Gold
Shaka = 100 for 1g
Drachme = 100 for 1g
Pound = 110 for 1g
Potato = 100 for 1g
Remember, King George still generates the same amount of gold as before, so he still only makes the 5gpt revenue he did before printing the money. Now he wants to buy some tech from the Zulu. Shaka wants 300 shaka, or 3g, for the tech. Before printing the excess money, George would have to pay 300 Pounds, which equalled 3g at the time. Now he has to pay 330 Pounds.
Now you are saying, why would a player purposely cause inflation. They probably would not unless forced too. Here is another example"
Shaka's citizens are not very happy, but her purse is tight. If she reduces tax rates she cannot afford to keep her empire running, or can she. SHe decides to print money to keep up with the reduced tax rate.
Gold
Shaka = 110 for 1g
Drachme = 100 for 1g
Pound = 110 for 1g
Potato = 100 for 1g
Now Shaka and George will trade on even terms, but if they deal with the others, they will be at a disadvantage.
Here is the final place where this really matters in terms of penalties.
Novgorod loses a war, badly. As part of the peace settlement, Novgorod has to pay a huge fine. Of course no one can do it at once, so its a yearly fee. Now they cannot support themselves and pay the fines. They are forced to print money to make minimum payments.
Gold
Shaka = 110 for 1g
Drachme = 100 for 1g
Pound = 110 for 1g
Potato = 160 for 1g
These exchange rates would also effect trade and trade deficits.
I'm not sure how to make your currency more valuable, suggestions are welcome.