Would it be possible to build a "rocket" using a series of ducted fans? I'm wondering if it would be possible to build ducted fan "rocket".

My idea is to construct the duct from 18650 batteries, and to have a series of contra rotating fans along its length.

I can 3d print new blades to have a steeper angles as the air-flow/speed increases.

This is the fan I'm looking at.

A 90mm fan will let you make a ring of 16 batteries around the curcumference and it would be 6 batteries tall as it's a 6s motor.

Is this feasable/will it work? what kind of thrust would I get?

Here is a quick render I made:
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.