Count the times a digit has appeared in a list as I scan the list

cnts = Transpose[{#, Module[{o = Ordering@#}, 
                      o[[o]] = Join @@ Range@Tally[#[[o]]][[All, 2]]; o]}] &;

Use (l containing desired list target):

result=cnts@l;

Will be order(s) of magnitude faster on large lists than OP method.

Here is a semi-imperative method:

runningCount[list_] := Module[{c}, c[_] = 0; {#, ++c[#]} & /@ list]

Example:

runningCount[{4, 2, 4, 0, 1, 0, 0, 2, 0, 0}]

(* {{4, 1}, {2, 1}, {4, 2}, {0, 1}, {1, 1}, {0, 2}, {0, 3}, {2, 2}, {0, 4}, {0, 5}} *)

I'll join the party :)

Clear["Global`*"]
lst = {4, 2, 4, 0, 1, 0, 0, 2, 0, 0}; 
Scan[(x[#] = 0) &, Union[lst]]; 
(Last@Reap@Scan[ Sow[{#, ++x[#]} ] &, lst])[[1]]

The idea is to set up a hash lookup counter of each number in the list, initially at zero. Then scan the list, incrementing the counter by one using lookup each time.

Timings

I did basic timings for the solutions given. all are using this list, and using AbsoluteTiming command

 lst = RandomInteger[10000, 50000];

Result

Ciao solution: 0.015831 seconds  
W Reach solution: 0.15155 seconds
Nasser solution: 0.22417 seconds
David Keith solution: 2.3196  seconds
A.G. solution:  145.95 seconds 

Code

Clear["Global`*"]
SeedRandom[1]
lst = RandomInteger[10000, 50000];
AbsoluteTiming[
 Scan[(x[#] = 0) &, Union[lst]];
 (Last@Reap@Scan[Sow[{#, ++x[#]}] &, lst])[[1]];
 ]

Clear["Global`*"]
SeedRandom[1]
lst = RandomInteger[10000, 50000];
AbsoluteTiming[
 Table[First@Tally@Reverse@Take[lst, i], {i, 1, Length@lst}];]

Clear["Global`*"]
SeedRandom[1]
lst = RandomInteger[10000, 50000];
counts[l_] := 
 Table[{l[[n]], Count[l[[1 ;; n]], l[[n]]]}, {n, Length[l]}]
AbsoluteTiming[counts[lst];]

Clear["Global`*"]
SeedRandom[1]
lst = RandomInteger[10000, 50000];
cnts = Transpose[{#, 
     Module[{o = Ordering@#}, 
      o[[o]] = Join @@ Range@Tally[#[[o]]][[All, 2]]; o]}] &;
AbsoluteTiming[cnts@lst;]

Clear["Global`*"]
SeedRandom[1]
lst = RandomInteger[10000, 50000];
runningCount[list_] := Module[{c}, c[_] = 0; {#, c[#] += 1} & /@ list]
AbsoluteTiming[runningCount[lst];]