I have a code structure that looks like this:
Class A:
  def __init__(self):
    processes = []
    for i in range(1000):
      p = Process(target=self.RunProces         
        There are a couple of syntax issues that I can see in your code:
args in Process expects a tuple, you pass an integer, please change line 5 to:
p = Process(target=self.RunProcess, args=(i,))
list.append is a method and arguments passed to it should be enclosed in (), not [], please change line 6 to:
processes.append(p)
As @qarma points out, its not good practice to start the processes in the class constructor. I would structure the code as follows (adapting your example):
import multiprocessing as mp
from time import sleep
class A(object):
    def __init__(self, *args, **kwargs):
        # do other stuff
        pass
    def do_something(self, i):
        sleep(0.2)
        print('%s * %s = %s' % (i, i, i*i))
    def run(self):
        processes = []
        for i in range(1000):
            p = mp.Process(target=self.do_something, args=(i,))
            processes.append(p)
        [x.start() for x in processes]
if __name__ == '__main__':
    a = A()
    a.run()
                                                                        A practical work-around is to break down your class, e.g. like this:
class A:
    def __init__(self, ...):
        pass
    def compute(self):
        procs = [Process(self.run, ...) for ... in ...]
        [p.start() for p in procs]
        [p.join() for p in procs]
    def run(self, ...):
        pass
pool = A(...)
pool.compute()
When you fork a process inside __init__, the class instance self may not be fully initialised anyway, thus it's odd to ask a subprocess to execute self.run, although technically, yes, it's possible.
If it's not that, then it sounds like an instance of this issue:
http://bugs.python.org/issue11240
It should simplify things for you to use a Pool.  As far as speed, starting up the processes does take time.  However, using a Pool as opposed to running njobs of Process should be as fast as you can get it to run with processes.  The default setting for a Pool (as used below) is to use the maximum number of processes available (i.e. the number of CPUs you have), and keep farming out new jobs to a worker as soon as a job completes.  You won't get njobs-way parallel, but you'll get as much parallelism that your CPUs can handle without oversubscribing your processors.  I'm using pathos, which has a fork of multiprocessing because it's a bit more robust than standard multiprocessing… and, well, I'm also the author.  But you could probably use multiprocessing for this.
>>> from pathos.multiprocessing import ProcessingPool as Pool
>>> class A(object):
...   def __init__(self, njobs=1000):
...     self.map = Pool().map
...     self.njobs = njobs
...     self.start()
...   def start(self):
...     self.result = self.map(self.RunProcess, range(self.njobs))
...     return self.result
...   def RunProcess(self, i):
...     return i*i
... 
>>> myA = A()
>>> myA.result[:11]
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
>>> myA.njobs = 3
>>> myA.start()  
[0, 1, 4]
It's a bit of an odd design to start the Pool inside of __init__. But if you want to do that, you have to get results from something like self.result… and you can use self.start for subsequent calls.
Get pathos here: https://github.com/uqfoundation