Both types OP amps should work fine with AC signals.
Try to build the simple type called "Ultra low distorsion amplifier" from this Data sheet https://www.google.se/url?sa=t&source=web&rct=j&url=https://www.bucek.name/pdf/lm381.pdf&ved=0ahUKEwiOkN_G3YbWAhVpJpoKHTttAnUQFgglMAA&usg=AFQjCNG75t6JtC-wSrtCMiMp3wJ7sKF33A
It will give you an amplifier with 10 times voltage gain.
If you replace the 100k resistor between input and output with a 1Mohm type your gain will go up to 100times.
I belive that you probably need to power the OP amp with more than 5V. 12V is more suitable.
Once you get this working you can start looking for low voltage OP amps if you want to power the amplifier from 5V.
As an alternative if you are not that comfortable working with analog circuits you can use the cicuit below that uses a single 74HC04 gate.
The feedback resistor forces the gate in to its analog region and creates an amplifier with lots of gain. In fact it has so much gain it will probably oscillate without input signal. But once you have a few millivolts of signal it will amplify it cleanly.
Should work well driving from 5V,