How do I download my Lightroom 5 to a new computer?

Or download the Trial version from Adobe

https://www.adobe.com/cfusion/tdrc/index.cfm?product=photoshop_lightroom&promoid&promoid=DTEML

Enter your serial number to license it.

if you follow all 7 steps you can dl a trial here:  http://prodesigntools.com/lightroom-5-ddl-comparison-vs-lr4.html

and activate with your serial.

if you have a problem dl'g, you didn't follow all 7 steps.  the most common error involves failing to meticulously follow steps 1,2 and/or 3 (which adds a cookie to your system enabling you to download the correct version from adobe.com).