In the beginning I also struggled with spectre requirement of newer openssl (1.1.0) than is in CentOS I have on VPS. Gandalf's advice to use older version which builds with openssl 1.0.1 helped me, unfortunately I can't find page he referred.
But basically it was this:
git clone https://github.com/spectrecoin/spectre
git checkout v1.3.5

I had to get over a few other issues, like install newer version of gcc (7.2.1 which supports C++14 standard - see RPM package devtools-7-runtime) and also build Boost C++ library (distro version didn't work, don't remember exact reason).

Finally I didn't know how to run it. Dunno is it is the best way, but I run it with -cli parameter under tmux console multiplexer.
When I ran it as daemon, I couldn't figure out how to connect to it to issue commands.
Also I needed to create .spectrecoin/spectrecoin.conf in my home dir where I enabled staking and set rpcpassword.
I wonder if there is a tutorial or comprehensive doc about this all (how to configure and run spectre without GUI). I didn't find any - I put everything together from pieces I googled out.