Getting compost from your local landfill is not a bad idea. However you should be using it as a soil conditioner and not as the total medium to grow your vegetables.
When starting a raised bed from scratch that is based on top of the garden soil, you deep dig the soil incorporating farmyard manure (fresh or bought in bags) so that it improves the soil structure and gives a good base for years to come. This also raises the level of the soil in your bed that is never walked on again.
Getting that base soil right will give your soil the body to hold water without getting water logged and also have the nutrients required for good crops.
In the following years there should be no need to deep dig, just incorporate some of that compost from the landfill into the top soil and a dusting of fertilizer.
You are right to think about the lime content. Your soil should be alkaline for the veg and would advise you to buy a cheap soil testing kit. Easy to use and will tell you the PH of your soil. You can then add lime as needed.
Growing potatoes on a new plot is a good way of "cleaning the soil" ready for other crops.
Also look at growing a green manure that is dug in to improve the soil.
Keep going, you will get there and it is well worth it.
Hope this helps.