I will assume that the base currency is C1 since that is the initial balance.

Day 1: 1 C1
Day 2: sell 1 C1 for 10 C2, no change in price, so 0% gain, still 1 C1
Day 3: price falls to 1 C1: 1 C2, so 900% gain, sell 10 C2 for 10 C1
Day 4: sell 10 C1 for 1,000 C2, no change in price, so 0% gain, still 10 C1
Day 5: price rises to 200 C1: 1 C2 from 100 C1: 1 C2, sell 1,000 C2 for 5 C1, so 50% loss, now 5 C1

To sum:

0% gain by 900% gain by 0% gain by 50% loss
= 1 * 10 * 1 * 0.5
= 5
= 400% overall gain

I can provide more detail if needed, but a base currency must be selected. The price only needs to be taken into account if there is a position still open; in other words, the price should be applied in this case if you still own something other than C1, the base currency.

If all positions are closed out before accounting for gains, only the trades need to be used to calculate the rate of return.

The simple formula for measuring relative prices is x_final/x_initial, and the change/return/yield is x_final/x_initial-1, with gain >= 0% and loss <= 0%.

Directly applying this formula to the trades, denominated in C1 is:

(10 C2/1 C1) * (10 C1/10 C2) = 10
(1000 C2/10 C1) * (5 C1/1000 C2) = 0.5

The overall relative prices is 5, expressed in percent is 400% because the percentage gain of 5 is (5-1)*100%.

If there is an open position, to account for change:

(# of C1 paid/# of C2 received) * (current price expressed as quotient of C2/C1)

Arithmetic vs geometric returns

I reread the final portion of your question. First, the calculations are incorrect, so you should follow my example.

Second, one cannot add returns, the total arithmetic return. They must be calculated geometrically, and that is why the each final/initial should be multiplied.

The simplest way to track profits and losses from trading is to mark your holdings to market in your local currency each day. That is, each day, value all your holdings in your local currency at their market price and sum them. If that number goes up, you made a profit on your combination of holding and trading currencies. If that number goes down, you took a loss.

There are various benchmarks you can compare that to. One is your total investment, which you should also value at market price as you invest it. So if you put in one Bitcoin when Bitcoins are worth $750, you add $750 to your total investment. Another is how much you would have if you had simply held Bitcoins instead of trading. Another possibility is to compare against other types of investments such as stocks or bonds.

Note that holding and trading crypto-currencies can be profitable but is extremely risky.

It really depends on what you call profit. If having more X means you made a profit, denominate in X!

If you want to compare two trades you generally want to see if you ended up with more than you went in with. Did a buy result in a better sale, or vise versa.

X can be loafs of bread, cars or preferably other currency.

Ultimately it is about increasing total wealth (I assume). That means you have to measure wealth. How you do that is up to you!

To be more direct: Pick either of those, or their value to a third currency. In the end what matters is how much you can buy with it, so picking a very stable third item is a smart move. If you care just about boosting these two numbers you can express your total value in one of the two.