You should do basic strength training using progressive overload, at a slower rate than someone who is not simultaneously working hard at karate. After you've built a base of strength through weight training, you should add some explosive movements so you can utilize your newfound strength to produce speed and power.
Basically, this means you should do basic whole-body barbell exercises like the squat and deadlift alongside some combination of fundamental upper-body exercises like bench press, overhead press, Turkish get-ups, chin-ups, pull-ups, and dips. (Don't do all of those; just pick a few.) Start light, practice good form, and add weight regularly. A basic program might be:
- Warm up
- 20 barbell squats
- Alternating maximum sets of pull-ups and dips, 5 times through with little rest
or, try Jim Wendler's advice for combat sports athletes:
Day 1: squat, bench, any other minor exercises you feel like doing
Day 2: deadlift, overhead press, any other minor exercises you feel like doing
or, if you refuse to get a barbell and squat rack:
- Warm up
- 10 reps (each side) of Turkish get-ups with a dumbbell
- Two sets of 20 dumbbell squats
- Pull-ups, three maximal sets
or use a vetted strength (not bodybuilding) program like Starting Strength (progressing slower than usual, and maybe only lifting twice a week), or 5/3/1, or GreySkull Linear Progression.
After you achieve a 1.5x bodyweight squat or 2x bodyweight deadlift, look into power variations of the Olympic lifts (power clean, push press, and so on) to develop explosiveness and speed. You'll get your conditioning from karate class; don't worry about cardio.
Don't use lifting as an excuse to skip class or to perform poorly in class. Lift after class, or on the day after class, so that you have plenty of time to recover before your next karate class. Practice kata and hitting things (heavy bags, trees) whenever you like, but don't let it interfere with class or lifting.