The most important areas on your body to strengthen before your Kilimanjaro climb are your legs and your core. Strong legs are essential as they are going to be doing most of the work, but a strong core will help keep pressure off your back as you climb carrying gear.
In addition to hiking and long walks, targeted muscle exercises such as squats, lunges and deadlifts are great for strengthening the legs.
Try to add and increase weight as you get comfortable with a set of up to 10-15 repetitions each. In addition, core strengthening exercises like pilates, planks, and abdominal work will support your back.