This is a much bigger project than you think, and processing power is the least of your concerns. You may want to skip straight to step #9. Otherwise, here are some questions you need to ask yourself, based on my experience of building something very similar as a hobby project (I used a desktop PC for processing):
1: Exactly how far should the robot reach? You say 1.25 meters - but at what height, and in what direction? If it absolutely must reach anything within a 1.25 meter dome around its base, then either it's maximum reach must be longer, or you have a very big engineering challenge. Look at some of the robots from Fanuc or Kuka; their online catalogs have cutout views that show how far the arm will reach in each direction, and those areas are never perfect domes. Also, how close to the base must the robot reach? If the first arm (from the base) is significantly longer than the 2nd arm, then it won't be able to reach anything close to the base. Also, 6 DOF DOESN'T MEAN IT CAN HAVE ANY POSITION WITH ANY MOVEMENT! The shoulder and elbow joints of robot arms are almost always in the same DOF (and sometimes the wrist too). You can move the arm in any direction or hold the grasper in any orientation, but you can't do both at the same time. If you look at the human arm, the shoulder has 3 DOF, the elbow has one, the forearm has one, and the wrist has two. That's 7 in total, and that's what you need for total flexibility. If you search for 7 DOF, you'll see examples showing how they can move in ways that 6 DOF arms can't. This may not be important for your project, but it's important to know because otherwise you may do a lot of needless debugging.
2: Start with the grasper first. There are companies that make these specially, and it's best to look at their work to see what you might need. These can be very expensive parts if they need motors or sensors. This is the most important part of the arm, and fortunately it's the first part you can prototype and test. Figure out what you want, cheaply and quickly fabricate a version yourself, and move it around by hand to ensure it actually does what you want it to do. You may find that it needs to do more or less than you expected. I've seen people learn a lot simply by using a pair of popsickle sticks.
3: Exactly how much force should the arm lift? Start with the outermost part, the end effector (aka grasper/gripper/whatever). That needs to have a grip capable of holding 2kg, or roughly 20 newton-meters (you'll be doing all forces in newton-meters now; it's a lot easier than foot-pounds because 1kg is 9.8NM, and it's best to round up). How does it hold it? Does it pinch sideways? If so, it needs to have some amount of grip. If it lifts from the bottom like a fork, it needs to have a certain amount of torque in the last up/down DOF. We'll come back to this.
4: (AKA #3 part 2) Now that you know what the end effector needs to do, you can get a better idea of how it should be shaped and how much it will weigh. And after going through step 1 and lots of graph paper, you may know how long each segment of the arm needs to be. With this knowledge, you can start calculating how much each segment will weigh and therefore how much force each DOF needs to support. It's safest to calculate this with each segment fully outstretched, but for the lowest 2 joints you may be able to estimate that they will always be at an angle and therefore don't need quite as much force.
5: Time to look at motors, the beautiful yet evil Medusa of robotics. Again, what does the robot need to do? Does it need to be fast, or powerful, or precise? If it needs precision but can be slow, you may need geared stepper motors. If it needs to be fast or powerful, you may need geared DC or AC motors. Speaking of which, DC motors will probably be easier to control. But each of these options have problems; stepper motors are quite weak and they don't know their position, while DC and AC motors simply don't know their positions. You'll need rotary encoders for DC/AC motors (incremental encoders at minimum), and may need absolute encoders no matter which motor type you choose. Absolute encoders tell you the exact angle of the arm, incremental ones will only tell you the change in angle. So, absolute encoders are better, but they are also much more expensive. You can avoid this problem by purchasing industrial servo motors (which look a lot like the stepper motors), but those are much more expensive because they have rotary encoders already built in. But those servos may be worth it just for the easier integration, if the maker has a nice software package that you can use.
6: You'll need to power all these motors (probably through drivers, which are another piece of hardware), and also power all the sensors and controllers, and they probably won't require the same voltage. Hopefully your motors can at least share a single voltage. You can buy velcro loops and labels in bulk.
7: Don't forget the weight of the motors in the arm; you'll need to include that when calculating the torque required for the next motor.
8: Make sure you have some arrangement for fast and cheap shipping of new components, and whatever discounts possible.
9: Realize this is way too complicated, and look up a robot refurbisher/supplier in your area. Retired industrial robots are much cheaper than new ones, and they can help you get set up and figure out what adjustments you need.
