Hey, 63/37 is a BAD alloy to use to put wires together especially for a first timer That solder doesn't flow period. You take the heat away and it solidifies. You really need to use 60/40. You got a solder with flux, so that is a good thing.
You also picked a non tin plated wire. Copper oxidizes and it won't solder well. If you have ever done plumbing and tried to solder copper tube without mechanically sanding the copper, it won't work.
The standard rules apply for soldering. Tip clean and tinned. Apply heat to the more massive part. Do not drip the solder onto the connection.
What can you do about it?
Take your cat 5 wire and do the following for practice:
Strip the end of two wires or a loop (your only practicing) 1" or so
Get a little vinegar and add some salt to it. Enough to dip the end of the wire in. The copper wire should come clean instantly. Rinse with water.
Cross the wires in the center and twist one toward the insulation of the other wire and do the same to the other one. The Western Union splice.
If you can suspend the splice. Heat from the bottom and apply solder along the length of the top of the wire.
The wire you really want is tin plated copper and the solder you really want id 60/40 assuming non-ROHS.
You should see a big difference if you use the Vinegar and salt technique or scrape a solid wire with sandpaper or a razor blade. A non-acid flux might be required, but really should not be. The solder could also be sub-par.
What I am saying is that you picked a difficult combination to solder for your first time. It's like trying to solder a 2" copper tube when you have never done a 1/2" copper tube. Silver soldering is an art too and so is say joining two pieces of quartz or borosilicate glass tubing together and so is welding (TIG, MIG and stick). All different and I've done them all. AT work, I was the only one that didn't have trouble silver soldering.