Using a sub with no sub out on my amp

So running my speaker cables out of the back of my amp into the sub is not going to change the sound that I'm getting out of my speakers is it going to alter the sound in any way obviously other than adding bass through the sub ?

You can also just run leads from the speaker to the sub. It won’t effect your sound unless you are a bat with super hearing. 
Not all subs take this type of connection. Rel is a brand that does. 

Id save more money,250 isnt going to get you anything worth buying .Get a used Rel t5i and run it high level output off your speaker outs

Thanks guys for the replies I'll look into that suggestion see what I can find one for.  Either that or I just look for another amp.

Your amp is fine. 2 options- buy a sub with speaker level inputs, or add a Line Out Converter to run a sub at line level off your speaker terminals. Either way no changes to the amp and won't affect the amp.

A LOC is just a couple 10k to 20k resistors that drops speaker output down to line level. They are under $20 at Parts Express, or you can even DIY for only about $2. That's what I did, see the pic on my system page. https://systems.audiogon.com/systems/8367?utm_campaign=website&utm_source=sendgrid&utm_mediu... If its good enough for me....  
With subs don't bother spending a lot on one it is better to use more, but one is better than none- 2 is better than 1, etc.

Don't worry about which channel either. Seriously, I have done it all these different ways, adding subs to an amp with no sub out is the easiest thing you can do.