Skill and technique are more important than anything in Tennis but by strengthening your body you will be able to run and shoot faster, so strength training is definitely important for this sport. Even a sprinting increase of a few milliseconds can make a difference and the harder your shots are the harder it will be for your opponents to keep up with you.
Upper Body Training
Probably the best strengthening exercises for tennis players are the ones which utilize a kettlebell. These exercises are a mix between a strength building exercise and a cardiovascular exercise.
The Single Arm Swing: This exercise will grant you more power and improve your serve. To execute this lift simply pick the bell up off of the ground with one arm. Next, rotate your fist so that your knuckles are pointing right at the sphere of the bell. From here on simply swing the bell between your legs and up to chest level. Make sure to properly utilize your hips in order to build the momentum to swing the kettlebell.
Kettlebell Long Cycle: You can do this with one or 2 kettlebells. Your choice. The Long Cycle is an exercise in which you must first lift (clean) the kettlebell to your chest and then perform a jerk with the kettlebell to an overhead lockout position. The clean and jerk is performed over and over again without setting the kettlebell down. Repetitions or length of duration depends on an individual’s choice or capability. In Kettlebell Sport competitions the competitors usually have 10 minutes to complete as many repetitions as possible as a judge counts and grades each lift.
These exercises are great for your wrist, elbow and rotator cuff strength. The stronger those muscles are the harder your shots will be. You can also do the deadlift to strengthen your back and core. The body works as a whole unit so this will help you out even more with your shots.
Lower Body Training
To increase running speed you should strengthen your legs with the squat and the leg extension exercise. There are several other leg strengthening exercises but from my experience those two are the best, especially the squat. Do heavy weight low rep training like a powerlifter does to strengthen the muscles as much as possible without gaining extra muscle which would slow you down.
A lean physique is best for the Tennis player.