How much carbon does a forest contain?
This question is very broad and this would require a lot of calculations, I will focus on the carbon dioxide in the trees, hopefully some other users can add details in respect to the organism and their carbon dioxide storage.

I found an article(R.Jandl et al.,(2007)) which did some calculations, this site does a really good job in explaning these calculations so I will cite that: 


  As trees photosynthesise they use carbon dioxide (CO2) from the
  atmosphere with water from rain or irrigation and nutrients from the
  soil to form carbohydrates, which make up the tree’s biomass, but how
  much carbon is made by a tree in this process? Researchers at
  Ecometrica have worked it out! The amount of carbon stored by a tree
  depends on its size, which in turn is influenced by factors, such as
  species, local environmental conditions and the way it is managed. In
  an attempt to find a simple answer to this question, researchers at
  Ecometrica have broken down approximately how much carbon is stored in
  each element of a typical tree (the branches, the leaves, the stem and
  the roots) by percentage for a quick and simple calculation. 
  
  They used standard forestry practices to estimate the amount of carbon
  contained within the stem, branches, roots and leaves of a mature
  sycamore (Acer pseudoplatanus), found near their office in Edinburgh,
  with a diameter of 52 cm and a stem height of 12 m.
  
  First they measured the various parts of the tree and, from these
  measurements; they calculated their volumes, which were then used to
  calculate the biomass (mass of living matter in plant tissues):  The
  radius of the stem at 1.3 m above the ground (r1 = 26 cm) and at the
  top of the stem (r2 = 20 cm), and The height of the stem (h = 12 m)
  They used known relationships between the biomass of the stem and the
  biomass of roots, branches and leaves to estimate the total biomass of
  the tree: Stem volume was estimated using the equation for the volume
  of a truncated cone: 
            
  
  
  which gave an estimated stem volume of 2.0 m3 The wood density of sycamore is approximately 620 kg/m3, so using the equation:
  
  Mass = Density × Volume  the stem biomass was 1243 kg, or
  approximately 1.2 tonnes  The biomass of the roots, branches and
  leaves of a sycamore tree are known to be around 26%, 11%, and 1% of
  the total biomass, respectively. These proportions were used to
  estimate the total biomass of the tree without having to dig up roots
  or cut down branches.  In summary, the sycamore tree had a stem with a
  biomass of 1.2 tonnes, roots of 0.5 tonnes, branches of 0.2 tonnes and
  leaves of 0.02 tonnes, giving a total biomass of 2 tonnes. 
  
  The carbon content of woody matter (stem, branches and roots) and that
  of leaves is approximately 50% of  their biomass. Multiplying the
  total biomass by the proportion of that biomass which is carbon gives
  an estimate of the total amount of carbon stored in the tree, which in
  this case was exactly 1 tonne! Each tonne of carbon equals to
  approximately 3.67 tonnes of CO2.


Then the further calulations really depends on the density of the trees in the forest (if you know the exact number let me know). There are a lot of atricles which gave some numbers the range is based on these articles somewhere around 30-400(and some even 1200 trees). I used the data from this article, which measured the optimal density for growing. I assumed that the climax forest is just naturally so I just picked 400 trees per acre (this is just guessing offcourse and I just generalized this for all trees, but I saw 400 in an other article):

Then we can just do some simple math for the carbon dioxide for the trees: 


  total CO2 for the trees = 400 x 3.67 = 1468 tonnes 
  


Let me know if someone can find the tree density of a climax forest