In order to meet all mutual dependencies the various components have to be compiled in a particular order which is summed up below.
Pre-requisites:
bash
libstatgrab
alsa-lib
libpulse
lm_sensors
libconfig
Muparser
upower
polkit-qt5
sudo
xorg-xmessage
libXss (libXScrnSaver)
libXcursor
libXcomposite
libxcb
libxkbcommon-x11
libdbusmenu-qt5
libfm
menu-cache
gtk-update-icon-cache (gtk+3)
hicolor-icon-theme
xdg-utils
xdg-user-dirs
oxygen-icons
openbox
libxslt

0 (1 package)

    lxqt-build-tools

I (2 package)

    libqtxdg
    lxqt-menu-data

II (4 packages)

    liblxqt
    libsysstat
    qtxdg-tools
    libfm-qt

III (5 packages)

    lxqt-themes
    lxqt-qtplugin
    obconf-qt
    pavucontrol-qt
    qps
    qtermwidget
    screengarb
IV (14 packages)

    lximage-qt
    lxqt-about
    lxqt-admin
    lxqt-archiver
    lxqt-config
    lxqt-globalkeys
    lxqt-notificationd
    lxqt-openssh-askpass
    lxqt-policykit
    lxqt-session
    lxqt-sudo
    pcmanfm-qt
    qterminal
    xdg-desktop-portal-lxqt

V (3 packages)
    lxqt-panel
    lxqt-powermanagement
    lxqt-runner

These groups must be processed in this order but the order of components within the each group does not matter.

All components are compiled using CMake and GNU make.
It is strongly advised to do the builds out of source tree.
At least on some operating systems like Arch Linux and Debian LXQt doesn't seem to start when CMake variable CMAKE_INSTALL_PREFIX which is explained below is not set to /usr which is something we may have to investigate. On the other hand this assignment will hardly ever do harm so it should be safe to recommend setting /usr in general for now.
Compiling all components individually

The repositories of all components maintained by the LXQt project as well as nearly all othes stated above come with a file README.md providing information how to install the respective component.
This information applies to compiling the archive files of regular releases as well.
The names of all components in the synopsis above are links to those repositories you may just follow.

The general approach is something like

mkdir build
cd build
cmake <path to sources> -DCMAKE_INSTALL_PREFIX=/usr
make
make install

Directory build should be outside the directory of the sources. cmake and make can and should be run by regular users, make install requires root privileges in order to write to the FHS. These can be acquired by sudo. This is depicted in section Scripts to build in one step and mandates an invocation sudo make install. But when all components are compiled on their own its probably more convenient to open an additional root session and invoke make install there.