How to replace sugar with dry fruit in a cake recipe?

This will definitely require some experimentation on your part. As a starting point, I recommend comparing the sugar content of the fruit to the sugar content of sugar. Below, I use dates as an example.

• 48 grams of dates (approximately 2 dates) contains 32 grams of sugar (source)
• 48 grams of sugar contains 48 grams of sugar (source: common sense)

So dates are 2/3 (67%) sugar by mass. This means it takes 1.5 times the mass of dates as sugar, to get the same amount of sugar. For example, in a recipe with 100g of sugar, substitute 150g of dates.

Making the dry fruit into a paste seems like a good way to get the sweetness evenly distributed through your batter. I suggest re-hydrating in hot water. Then puree the re-hydrated fruit, and incorporate it into the wet ingredients. Measure the amount of hot water you use, and reduce the amount of liquid in the cake recipe accordingly. Depending on the type of liquid used in your cake recipe, you may be able to use that liquid to re-hydrate the dry fruit. I know from experience that citrus juice works well for re-hydrating dry fruit; I'm not sure about milk or cream.

Of course the results will not be exactly the same as the original cake recipe. Different types of fruit will seem more or less sweet. The fruit paste will add flavor and effect the texture of the cake. Typically it will probably make the cake more moist and dense. This technique will not be suitable for cakes which are intended to be very light and fluffy, such as angel food cake or sponge cake.