How would I convert 0.093 calories per gram per degree celcius to joules per kilogram per degree celcius?

Both use Celsius so you don't need to worry about that. So what's left is calories/gram to joules/kilogram. 1000grams per kilogram so multiply .093 calories/gram * 1000 grams/kilogram. The grams cancel out and your left with 93 calories/kilogram. Now to convert calories to joules. 1 calorie = 4.184 joules. To convert it to joules, you multiply 93 calories/kilogram * 4.184 joules/calorie. The calories in both cancel out and your left with 389 joules per kilogram per degree Celsius (I rounded out to 3 digits cause that's what you started with). When converting units, just take it one unit at a time till you've converted everything you need.