Do I have any chances to move this lilac without killing it?
What is the best time of the year for this?
Any recommendations in terms of tools, procedure, root cutting, replanting?
The last two times I had to remove a Lilac I had to use a Bobcat or a heavy duty pickup with chains. Both plants had been in the ground for thirty years or more. There was no way you could transplant them. In fact even after dragging them out they popped up new shoots for the next three months.
However this looks to be a cultivar and may not be so vigorous. You could investigate with a spade and trench around the plant to see how thick the roots are.
You also want to ask yourself how much work you want to spend on a plant that you can buy for $39.99 list price at a nursery. If you are spending more than a few hours to do this then buy new.
Fall is the best time to do this. Water the plant well for a few days before to loosen up the soil. Be prepared for a deep root ball which means you will be digging down 2 feet (60 cm). This can make a huge mess in your lawn so have some bags of top soil to fill the hole.
Have your destination hole ready to go before you start and plant it at the same level as it currently sits at. No need to add bonemeal or amend the local soil in any way.
A sharpened spade, secateurs and lopper are what you need if the plant is not too established. For well rooted plants removal is the only option with a pickup truck and chains. Safe operating practices are the only way to go! Make sure you are wearing safety boots and gloves, have a good strong chain, keep bystanders back some distance.
In your comments you mention that the plant is old with a diameter of 6' (2 M). I don't think you can successfully move this without a tree spade as the roots will go down quite deep. Better off to buy a new one.