CJam, 20 bytes

''S5N'5S5N'NS5N'SS5N

This approach doesn't use any built-in character counting.

Try it online in the CJam interpreter.

How it works

''S5N e# Push a single quote, a space, the integer 5 and a linefeed.
'5S5N e# Push the character 5, a space, the integer 5 and a linefeed.
'NS5N e# Push the character N, a space, the integer 5 and a linefeed.
'SS5N e# Push the character S, a space, the integer 5 and a linefeed.

CJam, 20 bytes

{`"_~"+$e`{)S@N}%}_~

How it works

We first start off with one of the standard quine in CJam

{`"_~"}_~

which pushes the first block on stack, copies it, and runs the copy, which makes it print the source code itself finally.

Then we add the logic to compute the character count from the source code:

{`"_~"+                         e# At this point, we have the full source code with us
       $e`                      e# Sort to get similar characters together and run RLE to
                                e# get count of each character as [count char] array
          {    }%               e# Run each array element through this loop
           )S@N                 e# Pop the character, put a space, rotate the count after
                                e# space and then finally put a newline after the trio
                 }_~            e# Second half of the standard quine explained above

Try it online here

CJam, 14 bytes

{S2N`/}`{S2N}/

Try it here.

Output is in the order they firstly appears:

{ 2
S 2
2 2
N 2
` 2
/ 2
} 2

It simply appends <SP>2<NL> to each character in {S2N`/}.

Python 3.5.0b1, 107 73 bytes

s="t='s=%r;exec(s)'%s;[print(c,t.count(c))for c in sorted({*t})]";exec(s)

Rather than the usual string replacement quine, which requires writing everything twice, here's an exec quine.

Mathematica, 101 bytes

Apply[Print[#1, " ", #2] &, Tally[Characters[StringJoin[ToString[#0, InputForm], "[];"]]], {1}] & [];

Unfortunately, I can't use any of the normal golfing tricks like removing whitespace, <> for StringJoin, # instead of #1, @ for prefix function calls or @@@ instead of Apply[...,{1}], because ToString[...,InputForm] thinks it has to pretty print everything...

This prints the characters in the order they first appear in the code. If I can assume that this isn't run in a REPL environment (which is rather unusual for Mathematica) I can save two bytes by omitting the two ;.

Haskell, 178 bytes

main=putStr(unlines[s:' ':show t|(s,t)<-zip" \"'(),-0123456789:<=S[\\]aehilmnoprstuwz|"[3,3,3,3,3,41,4,1,6,19,12,5,5,2,2,2,2,3,2,2,2,3,3,3,2,2,2,4,2,2,4,2,3,2,5,5,3,2,2,2]])--178

Nothing fancy. All characters of the program are in a literal list (String). So are the frequencies. Zip both lists and print. Output:

  3
" 3
' 3
( 3
) 3
, 41
- 4
0 1
1 6
2 19
3 12
4 5
5 5
6 2
7 2
8 2
9 2
: 3
< 2
= 2
S 2
[ 3
\ 3
] 3
a 2
e 2
h 2
i 4
l 2
m 2
n 4
o 2
p 3
r 2
s 5
t 5
u 3
w 2
z 2
| 2 

Dart - 214 127

A direct version:

main(){print("  22\n\" 3\n( 3\n) 3\n1 3\n2 15\n3 8\n4 1\n5 2\n8 2\n; 2\n\\ 23\na 2\ni 3\nm 2\nn 23\np 2\nr 2\nt 2\n{ 2\n} 2");}

The "4" is just a fiddling factor to make the numbers add up. See/run on DartPad.

Original: Standard quine tactic, and Dart's function names are a little too long for good golfing.

main({m,v,q:r'''main({m,v,q:r''}'')''{m={};for(v in q.split(''))m[v]=m[v]==null?2:m[v]+2;m.forEach((k,v)=>print("$k $v"));}'''}){m={};for(v in q.split(''))m[v]=m[v]==null?2:m[v]+2;m.forEach((k,v)=>print("$k $v"));}

See/run it on DartPad.