You can get the formula from the generic Barometric formula with lapse rate:
https://en.wikipedia.org/wiki/Barometric_formula
But don't fret about the large equation, for temperature correction you can assume most stuff to be constant.
You assume p1=p2 (because that is what your altimeter converts to an altitude) and then you can throw out the exponent and a lot of other stuff.
You end up with this equation:
h1/t1 = h2/t2
This assumes temperatures in Kelvin. For temperatures in C and solving for h2 the formula becomes:
h2 = h1 * ((273.15+t1)/(273.15+t2))
Your example:
h1 = 5000 ft
t1 = 15 C (standard ISA)
t2 = 25 C
result:
h2 = 5173 ft
t2 = 30 C results in h2 = 5260 ft
t2 = 35 C and h1 = 8000 ft results in h2 = 8555 ft
h1 is the altitude your altimeter is reading (it assumes ISA, so t1 = 15C)
h2 is the actual altitude you are at assuming a sea level temperature of t2.