Hey,
If I' not mistaken:
1. The turntable needs to go through the dj-mixer to get boosted to line-level.
2. Both samplers are already line-level, so they don't need to go through the dj-mixer.
With this in mind:
1. Once the turntable has gone through the mixer, it is at a line-level and the Akai should be able to sample it nice and loud. Make sure that the inputs of the Akai are set to line-level as well.
2. Once sampled, the Akai should play back the sample nice and loud if it is plugged directly into the amp.
3. If it is loud, you need to make sure that the inputs of the ASR are set to line-level. Plug the Akai into the ASR's inputs and sample.
4. When you plug the ASR back into the amp, the sample should be nice and loud.
Most samplers I've worked with, even older ones like the Emax II, have some sort of input metering which tells you how hot the input signal is before you sample it. See if your samplers have that feature, that way you can see at what point the sample is losing volume.
Take care,
Nick