In order to increase the chances of breaking even and making profit (starting to become slim) you will want to go cheap on some parts while you should never go cheap on others.
Don't skimp on the power supply! Seriously.
You don't need a SSD or HD! You can run a mining tailored OS off a thumb drive!
Go cheap on the CPU if you are doing GPU mining. It won't help. RAM is also not that important.
The heart of your OS will be the GPU (or GPUs). Lots of great cards out there but they are grabbed up quickly. Be patient and look for deals. You might end up paying an extra hundred bucks for each card if you don't. You will make more money by waiting for a good deal than you would mining during that period. I like the RX series but others love the 1070 and 1080s. Depending on what you are mining there actually won't be too much of a difference between 8gb and 4gb - other things such as GPU clock speed are more important. You should be able to compare the cards online with respect to mining a popular coin such as Eth.
Open air cases for mining are nice. If you are using PCI risers consider picking up a couple spares as they are cheap.
Make sure you buy the right motherboard in terms of compatibility.
https://pcpartpicker.com/ will help you verify if things are compatible if you cannot figure it out by reading the specs.
600$ will get you a rig with one or two relatively weak GPUs at the most. Make sure to consider extendability if you want to add more cards without having to pay for a whole new rig. A lot of the costs are the GPUs but a lot of the costs are also the rest of the rig.
And have fun.
That is all I can think of for now.