The simple approach would be to solve this with a CROSS JOIN like demonstrated by @jpw. However, there are some hidden problems:
The performance of an unconditional CROSS JOIN deteriorates quickly with growing number of rows. The total number of rows is multiplied by the number of weeks you are testing for, before this huge derived table can be processed in the aggregation. Indexes can't help.
 
Starting weeks with January 1st leads to inconsistencies. ISO weeks might be an alternative. See below.
 
All of the following queries make heavy use of an index on exam_date. Be sure to have one.
Only join to relevant rows
Should be much faster:
SELECT d.day, d.thisyr
     , count(t.exam_date) AS lastyr
FROM  (
   SELECT d.day::date, (d.day - '1 year'::interval)::date AS day0  -- for 2nd join
        , count(t.exam_date) AS thisyr
   FROM   generate_series('2013-01-01'::date
                        , '2013-01-31'::date  -- last week overlaps with Feb.
                        , '7 days'::interval) d(day)  -- returns timestamp
   LEFT   JOIN tbl t ON t.exam_date >= d.day::date
                    AND t.exam_date <  d.day::date + 7
   GROUP  BY d.day
   ) d
LEFT   JOIN tbl t ON t.exam_date >= d.day0      -- repeat with last year
                 AND t.exam_date <  d.day0 + 7
GROUP  BY d.day, d.thisyr
ORDER  BY d.day;
This is with weeks starting from Jan. 1st like in your original. As commented, this produces a couple of inconsistencies: Weeks start on a different day each year and since we cut off at the end of the year, the last week of the year consists of just 1 or 2 days (leap year).
The same with ISO weeks
Depending on requirements, consider ISO weeks instead, which start on Mondays and always span 7 days. But they cross the border between years. Per documentation on EXTRACT():
week
The number of the week of the year that the day is in. By definition (ISO 8601), weeks start on Mondays and the first week of a
  year contains January 4 of that year. In other words, the first
  Thursday of a year is in week 1 of that year.
In the ISO definition, it is possible for early-January dates to be part of the 52nd or 53rd week of the previous year, and for
  late-December dates to be part of the first week of the next year. For
  example, 2005-01-01 is part of the 53rd week of year 2004, and
  2006-01-01 is part of the 52nd week of year 2005, while 2012-12-31 is
  part of the first week of 2013. It's recommended to use the isoyear
  field together with week to get consistent results.
Above query rewritten with ISO weeks:
SELECT w AS isoweek
     , day::text  AS thisyr_monday, thisyr_ct
     , day0::text AS lastyr_monday, count(t.exam_date) AS lastyr_ct
FROM  (
   SELECT w, day
        , date_trunc('week', '2012-01-04'::date)::date + 7 * w AS day0
        , count(t.exam_date) AS thisyr_ct
   FROM  (
      SELECT w
           , date_trunc('week', '2013-01-04'::date)::date + 7 * w AS day
      FROM   generate_series(0, 4) w
      ) d
   LEFT   JOIN tbl t ON t.exam_date >= d.day
                    AND t.exam_date <  d.day + 7
   GROUP  BY d.w, d.day
   ) d
LEFT   JOIN tbl t ON t.exam_date >= d.day0     -- repeat with last year
                 AND t.exam_date <  d.day0 + 7
GROUP  BY d.w, d.day, d.day0, d.thisyr_ct
ORDER  BY d.w, d.day;
January 4th is always in the first ISO week of the year. So this expression gets the date of Monday of the first ISO week of the given year:
date_trunc('week', '2012-01-04'::date)::date
Since ISO weeks coincide with the week numbers returned by EXTRACT(), we can simplify the query. First, a short and simple form:
SELECT w AS isoweek
     , COALESCE(thisyr_ct, 0) AS thisyr_ct
     , COALESCE(lastyr_ct, 0) AS lastyr_ct
FROM   generate_series(1, 5) w
LEFT   JOIN (
   SELECT EXTRACT(week FROM exam_date)::int AS w, count(*) AS thisyr_ct
   FROM   tbl
   WHERE  EXTRACT(isoyear FROM exam_date)::int = 2013
   GROUP  BY 1
   ) t13  USING (w)
LEFT   JOIN (
   SELECT EXTRACT(week FROM exam_date)::int AS w, count(*) AS lastyr_ct
   FROM   tbl
   WHERE  EXTRACT(isoyear FROM exam_date)::int = 2012
   GROUP  BY 1
   ) t12  USING (w);
Optimized query
The same with more details and optimized for performance
WITH params AS (          -- enter parameters here, once 
   SELECT date_trunc('week', '2012-01-04'::date)::date AS last_start
        , date_trunc('week', '2013-01-04'::date)::date AS this_start
        , date_trunc('week', '2014-01-04'::date)::date AS next_start
        , 1 AS week_1
        , 5 AS week_n     -- show weeks 1 - 5
   )
SELECT w.w AS isoweek
     , p.this_start + 7 * (w - 1) AS thisyr_monday
     , COALESCE(t13.ct, 0) AS thisyr_ct
     , p.last_start + 7 * (w - 1) AS lastyr_monday
     , COALESCE(t12.ct, 0) AS lastyr_ct
FROM params p
   , generate_series(p.week_1, p.week_n) w(w)
LEFT   JOIN (
   SELECT EXTRACT(week FROM t.exam_date)::int AS w, count(*) AS ct
   FROM   tbl t, params p
   WHERE  t.exam_date >= p.this_start      -- only relevant dates
   AND    t.exam_date <  p.this_start + 7 * (p.week_n - p.week_1 + 1)::int
-- AND    t.exam_date <  p.next_start      -- don't cross over into next year
   GROUP  BY 1
   ) t13  USING (w)
LEFT   JOIN (                              -- same for last year
   SELECT EXTRACT(week FROM t.exam_date)::int AS w, count(*) AS ct
   FROM   tbl t, params p
   WHERE  t.exam_date >= p.last_start
   AND    t.exam_date <  p.last_start + 7 * (p.week_n - p.week_1 + 1)::int
-- AND    t.exam_date <  p.this_start
   GROUP  BY 1
   ) t12  USING (w);
This should be very fast with index support and can easily be adapted to intervals of choice.
The implicit JOIN LATERAL for generate_series() in the last query requires Postgres 9.3.
SQL Fiddle.