Where should I place the constant?
I will answer your question in an unexpected way : the better solution is, as far as I can tell, much more straightforward than what you are doing.

Indeed, you don't need a class : your class has an __init__ and a single method to_rna, it might as well be a single function.

from string import maketrans

def dna_to_rna(dna_string):
    """Return the RNA complement of the DNA strand."""
    return dna_string.translate(maketrans('GCTA', 'CGAU'))


I think see this the neatest solution. Then if on only want to create the table only once for performance reason, you can put it a variable outside the function but it's probably not needed.