If you just need to total number of 1's and don't need to keep the individual observations, you might consider the following which is a lot faster:

Total[RandomChoice[{p, 1 - p} -> {1, 0}, nSimulations]] // Timing
(* {0.046875, 699861} *)

And in that case even faster is the following:

RandomVariate[BinomialDistribution[z, p], 1] // Timing
(* {0., {700138}} *)

Would you elaborate on how you tested the speed? I'm assuming it must be in how you generated multiple samples from

If[1 <= RandomInteger[{1, n}] <= m, c = 1, c = 0];

I get the following:

nSimulations = 10000000;
n = 10;
m = 7;
p = m/n;(*Probability of observing a 1 *)

Timing[x = Table[If[1 <= RandomInteger[{1, n}] <= m, c = 1, c = 0], {i, nSimulations}];]
(* {10.140625`,Null} *)

Timing[x = RandomChoice[{p, 1 - p} -> {1, 0}, nSimulations];]
(* {0.171875`,Null} *)

Timing[x = RandomVariate[BernoulliDistribution[p], nSimulations];]
(* {0.171875`,Null} *)

Here are two ways to get a list of 0's and 1's with a specified probability:

p = 0.375;  (* Probability of observing a 1 *)
n = 10;  (* Number of random samples *)

x = RandomChoice[{p, 1 - p} -> {1, 0}, n]
(* {0,0,0,0,1,0,1,1,0,0} *)

x = RandomVariate[BernoulliDistribution[p], n]
(* {0,0,1,0,1,1,1,0,0,0} *)

In[7]:= f[prob_, n_Integer, seed_Integer] :=
 Block[{}, SeedRandom[seed]; 
  Boole[# <= prob] & /@ RandomReal[{0, 1}, n]]

In[9]:= f[.3, 20, 0]

Out[9]= {0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0}

In[10]:= f[.3, 20, 1]

Out[10]= {0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1}