It's clear you need to read the input in as a string (which you should
always do, no matter what - it prevents program breaking). Remember, pseudocode is one of the most useful tools you have when programming. For example:
variables:
string: input
double: degreesC
double: degreesF
start loop
ask for input
if input is 'Q' or 'q' then break out of loop
(verify input)
loop through the input string, making sure every character is a number or a dot, and the last character is either a 'C' or an 'F'.
if input ends in 'C':
copy number portion of input into degreesC
calculate and print degreesF
else if input ends in 'F':
copy number portion of input into degreesF
calculate and print degreesC
end loop
With something like that, you can easily see what you need to do.