Start it soon, and in small chunks. For all of May, devote 20 minutes a day to beginning the process: throwing out or donating unwanted items, acquiring packing materials, and grouping things logically.
For all of June, devote 20 minutes a day to packing. That may be one box or less, but make sure you’re doing it. Focus on packing away less-used items. (Don’t forget to CLEARLY mark any boxes with not only the room, but the contents of the box.)Store the boxes in a fairly out-of-the-way location, but one that’s easy enough to access when necessary.
By the beginning of July, the only things not packed should be the things you use regularly. In your daily 20 minutes, prepare the boxes for these things. Have them assembled and labeled, ready to pack. As moving day gets closer, pack away the last items, and marvel at how not stressed out you are.