Theoretical p-Phenylenediamine synthesis
I'd say the suggested reaction of benzene-1,4-diol (hydroquinone) with $\ce{PCl3}$ to get 1,4-dichlorobenzene would not work (see  Waylander's comment). However, you can buy 1,4-dichlorobenzene very cheap (e.g., USD 21.50 for $\pu{100 g}$ bottle from Sigma-Aldrich) so why you need this trouble?

The suggested reaction of 1,4-dichlorobenzene with $\ce{NH3}$ to get benzene-1,4-diamine (p-phenylenediamine) would work, however, because there is a literature evidence for that (Ref.1). To do so, you need to use copper(I) chloride catalyst ($\ce{CuCl}$) and use 28% $\ce{NH4OH}$ solution as $\ce{NH3}$ source. Also need to heat the reaction mixture above $\pu{200 ^\circ C}$ in a pressure bomb (you may need to read this reference carefully before you start the reaction).

References:


Armand J. Quick, "The Preparation of p-Phenylenediamine and Aniline from their Corresponding Chlorobenzenes," J. Am. Chem. Soc. 1920, 42(5), 1033-1042 (https://doi.org/10.1021/ja01450a018).