The process of installing a custom webpart solution doesn’t have to be a complex task. In fact all you need to do is run two commands In Power Shell and activate the custom webpart and you are done.

So here are some steps I took recently:
Copied the webpart to the local disk of my SharePoint server, I placed this file in c:install
Add-SPSolution c:installSharePointWebpart.wsp
Install-SPSolution –Identity SharePointWebpart.wsp –WebApplication http://somesite –GACDeployment

After the install of the w webpart you will need to do the following to make it active.
I navigated to the site where my new webpart solution was deployed: http://somesite
Then clicked on Site Action followed by Site Settings
Under Site Collection Administration I then entered Site Collection Features
Under Site Collection Administration > Features I located the webpart and clicked activate

Now the web part is available for use.

I hope this helps