Would it be possible to build a "rocket" using a series of ducted fans?
From the link you provided:


  Specification:
  
  
  Rotor Diameter: 90mm
  Fan blades:11 blades
  Weight: about 350g
  Working Voltage:6s(22.2V) lipo battery
  Motor:Brushless Motor 3553 1450kv
  No Load Current: 4.1 A
  Load Current: 83A
  No Load Speed:32190 rpm
  Load Speed:16095 rpm
  Thrust: 3300g
  G/A:45.16
  


Assuming that 16 * 6/4 = 24 18650 cells would be able to deliver full electrical power for the fan, the issue would be indeed be the local angle of attack of the fan blades.

Thrust at full power is listed at 3.3 kg = 32 N. This would be at standstill/hovering conditions at sea level, since measuring at that level provides the highest thrust level for advertisements. Diameter is 0.09m. Net thrust T = 

$$ T = \dot{m} \cdot (V_{out} - V_{in}) = \dot{m} \cdot \Delta V \tag{1}$$

$$ \dot{m} = \rho \cdot A \cdot V \tag{2} $$

Combining (1) and (2) for the hover, with $V_{in}$ = 0: 

$$ V = \sqrt{\frac{T}{\rho \cdot A}} = \sqrt{\frac{32}{1.225 \cdot \pi/4\cdot0.09^2}} = ~\text{64 m/s}$$

Impulse thrust considerations usually draw a contracting propeller wake for inducted propellers. Ducted fans work a bit differently and we can take the average velocity behind the fan for further Order Of Magnituding. Below figure is from this research paper, and shows the considerations for ducted fan flow; it contains some methods for more detailed computations. 



Rotational speed under load is 16095 rpm = 1,684 rad/s, tip speed = 0.045 * 1,684 = 75.8 m/s. A velocity triangle at the blade tip has as angle $ tan^{-1} (64/75.8) = 40 $ deg. The blade needs to be inclined further than that, usually about 6 deg, so tip blade angle of the standard fan would be 46 deg. Purchasing the actual fan for verifying the above would be a good thing!

For the second fan, this same method can be followed: mass stream will remain the same if the hull is closed, in order for the 2nd fan to deliver the same thrust $\Delta V$ = 64 => ${V_{}out} = 64 + 64 = 128 $ m/s. Tip angle velocity triangle = $ tan^{-1} (128/75.8) = 59.4 $ deg, fan blade angle = 59.4 + 6 = 66 deg, etc.

Note that the above is valid for the hover. As soon as the "rocket" picks up speed, the local angle of attack of the fan blades will reduce and therefore thrust will reduce. So one would have to optimise time of thrust (time of amps delivered) with weight, momentary speed, and expected end speed, then average the blade angles out for the speed function. 

Note that opening the hull in between the fans allows for extra air to be drawn in, increasing the mass flow. The paper cited above has results for a setup like that as well - if the increased mass flow relates in lower entry velocity, this might be worth considering.