If a PC wants to make a golem specifically, the various Manuals of Golems have them covered. That's pretty clear in the rules (Dungeon Master's Guide, page 180).
However, there don't appear to be many rules for how a PC can create constructs other than golems, such as animated suits of armor. (Page 129 of the Dungeon Master's Guide talks about how much material, time, and money is needed for magic items of known ratings – common, uncommon, rare, very rare, and legendary. However it is not clear whether animated suits of armor are "common," and possibly each Dungeon Master is supposed to make a campaign-specific ruling.)
(Animate Object lasts only 1 minute. Obviously a construct should be more-or-less permanent, and I'm used to how other editions handle converting temporary effects into permanent ones via a permanency spell or something similar, but permanency doesn't appear to be in 5e.)
So how is someone supposed to create constructs other than golems? The Animated Objects entry in the Monster Manual (p. 19) says:
Animated objects are crafted with potent magic to follow the commands of their creators.
Where are the rules for this potent magic? Does the game just leave it up to DMs to to define this "potent magic" via homebrewed rituals, new Manuals types, and whatnot?