Q:

How to update HPLIP in ubuntu 16.04?

My printer (hp oj250 mobile) requires hplip 3.16.8, but 3.16.3 is what I got when I installed hplip-gui on my freshly installed 16.04 box.
This is the response from the commands given in the comments to fix the dependencies:
$ sudo apt update&&sudo apt upgrade
[sudo] password for user: 
Hit:1 http://ppa.launchpad.net/deluge-team/ppa/ubuntu xenial InRelease         
Hit:2 http://archive.ubuntu.com/ubuntu xenial InRelease                        
Get:3 http://archive.ubuntu.com/ubuntu xenial-security InRelease [102 kB]
Ign:4 http://www.openprinting.org/download/printdriver/debian lsb3.2 InRelease 
Hit:5 http://www.openprinting.org/download/printdriver/debian lsb3.2 Release   
Fetched 102 kB in 2s (46.4 kB/s)                              
Reading package lists... Done
Building dependency tree       
Reading state information... Done
All packages are up to date.
W: http://www.openprinting.org/download/printdriver/debian/dists/lsb3.2/Release.gpg: Signature by key F8897B6F00075648E248B7EC24CBF5474CFD1E2F uses weak digest algorithm (SHA1)
Reading package lists... Done
Building dependency tree       
Reading state information... Done
Calculating upgrade... Done
The following packages were automatically installed and are no longer required:
  linux-headers-4.10.0-28 linux-headers-4.10.0-28-generic
  linux-image-4.10.0-28-generic linux-image-extra-4.10.0-28-generic
Use 'sudo apt autoremove' to remove them.
0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.

$ sudo apt install -f
Reading package lists... Done
Building dependency tree       
Reading state information... Done
The following packages were automatically installed and are no longer required:
  linux-headers-4.10.0-28 linux-headers-4.10.0-28-generic
  linux-image-4.10.0-28-generic linux-image-extra-4.10.0-28-generic
Use 'sudo apt autoremove' to remove them.
0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.

A:

You update the installed packages in Ubuntu by running sudo apt update&&sudo apt upgrade.  This will upgrade to the latest in the repository, which currently 3.16.3, what you already have.
When you need something more recent than what is provided in the repository you would have to go to a different source, in this case you can go directly to HP.
The latest HP provided drivers are located at: https://developers.hp.com/hp-linux-imaging-and-printing
That is a permanent link where they provide their latest drivers, which currently is version 3.17.11.
After you have run the install program, followup with the apt command to fix the missing dependencies.
$ sudo apt install -f

If you still run into problems with the dependencies make sure you have an updated repository cache and your libraries are also updated.  You can perform this with these commands:
$ sudo apt update
$ sudo apt upgrade

Run the HP Device Manager (HP Toolbox) from the Ubuntu Launcher.  If it doesn't automatically see your printer, you may have to apply Manual Discovery by supplying the IP address.
Manually adding the printer
From the Hp Device Manager click:
(You may have to run hp-setup from the commandline for the proper GUI interface.)

1) (the plus sign) `+` -> **Show Advance Options** ->
2) (check mark) **Manual Discovery** -> (typein the IP address of the printer) ->
3) (click) **Next** -> (Select the desired printer) ->
4) (click) **Next**
5) Follow the GUI as you normal

Final results:
From the status of the chat messages, the OP didn't have any of the repository update entries in his apt configuration.  The HP screen mentions that as a prerequisite.
The missing entries were:
deb us.archive.ubuntu.com/ubuntu xenial-updates main restricted
deb us.archive.ubuntu.com/ubuntu xenial-updates universe
deb us.archive.ubuntu.com/ubuntu xenial-updates multiverse

