Reaction of citric acid and calcium chloride
This procedure is done in our laboratory class, but first you have to add sodium hydroxide until the lemon juice is somewhat alkaline. Then add the calcium chloride (use $1~\mathrm{ml}$ for every $2~\mathrm{ml}$ of the lemon juice + $\ce{NaOH}$ solution).
I think the calcium chloride would first react with the hydroxide, producing calcium hydroxide, which would then react with citric acid to form calcium citrate.

$$\ce{2 C6H8O7 + 3 Ca(OH)2 -&gt; Ca3(C6H5O7)2.4H2O + 2H2O}$$

Filter out the salt (precipitate) formed, weigh it, then you would have to add sulfuric acid afterwards to react with the calcium citrate. This would then form citric acid (soluble) and a precipitate which is calcium sulfate. 

$$\ce{Ca3(C6H5O7)2.4H2O + 3H2SO4 -&gt; 2C6H8O7 + 3CaSO4.2H2O + 2H2O}$$

You would then have to concentrate the filtrate to obtain the citric acid crystals.