oiche said:
The comments about placement are well advised. I also have tight space issues and an even tighter budget. I have a project genie mk3 deck, marantz 6002 amp, dali lektors (on stands) with decent connectors and it sounds bright and clear. If you make the best of what you have with some practical arrangement you can achieve decent enough results.
The 'arrangements' are very much the key here. Dali Lectors on stands will be pretty tight and a hugely prefereable to the OP's proposed setup and this in itself will make a big difference. The issue with the OP's system is the transmission of low frequency enargy through the furniture and structure of the room and this will muddy up the sound.
To solve this you need to isolate the source of the energy, the speakers, from the player. Proper speaker stands are the way to do this but if they are not possible then you need to find a way of 'de-coupling' the speakers from the room. Remember, the frequencies you need to worry about are low frequencies so soft or resilient feet on the speakers are unlikely to do the job, best bet would be spikes or cones, downward facing onto the furniture.
Ideally the record player should be wall mounted, a purpose built stand is best, the Apollo being the cheapest effective one I know but there are others, some rather better, some not. If that is out of the question, then see if you can get an isolation platform, again spiked to the chest of drawers,
This might sound excessive but in my experience makes a big difference, it is up to you of course, advice is free, make of it what you will.