Use a numerical method of your choice eg Runge Kutta (or even Euler) plus the Shooting method. The shooting method converts a boundary value problem to an initial value problem.

see https://www.youtube.com/watch?v=ZMgikZ-lcS8

https://www.youtube.com/watch?v=6ONtZz0fmvw&list=PLC245A479FD6059DB

https://www.youtube.com/watch?v=UmSVUqkKvrE&list=PLC245A479FD6059DB&index=2

1) Guess an initial value for F''(0) as you don't know what the actual value is.

2) Run the Runge Kutta or whatever method [using the initial conditions F(0)=F'(0)=0 and guessed or interpolated value of F''(0)] to a high value of t, which you consider to be infinity and call this value of time T. So T is an approximation to infinity.

3) Check the value of F''(T) for how close it is to one

4) If F''(T) is close enough to one you have a good enough solution so stop, otherwise continue to step 5)

5) If you have two old values for F''(0) use interpolation to choose a better value for F''(0) and go to step 2). If you only have one old value for F''(0) guess another value for F''(0) and go to step 2)

for interpolation

BetterValueF''(0)=F''(0)oldold+(( F''(0)old- F''(0)oldold)/(F''(T)old-F''(T)oldold))*(DesiredF''(infinity)-F''(T)oldold)

where DesiredF''(infinity)=1

Hence, in time F''(0) changes to a value which when used in the Runge Kutta or whatever method, gives you the necessary boundary conditions.

this is an autonomous equation.

Handbook of Exact Solutions for Ordinary Differential Equations

in this book page 660 number 9 is your answer

The ODE that you have posted looks very much like the plane stagnation flow problem also called Hiemenz flow. However the BC at infinity is incorrect. You state F''[infinity)=1 whereas in Hiemenz flow it is F'[infinity]=1.  In any event either problem can be solved  numerically by a shooting method. Of course you do not need to specify F'''. Since your ODE is homogeneous and is of order 3 you need to specify 3 boundary conditions. I have attached some notes that show how to solve this problem using a shooting technique using Mathematica. Very simple!

You may diminish its order by substitution F'=p and recalculating the BC.This result in as follows d2p/dF2=1-1/p2-Fp'/p-(p'/p)2,  where p'=dp/dF. 

Use the particular  solution F_0(t)= t    for reduction of Your equation to the second order nonlinear ODE.