Changing bicycle tire - problem

[Trixanity]

A relative had a tire that continued to go flat on every trip and after a couple patches and the tube finally snaking around the wheel which I had to cut to bits to get off, I decided to (well, had no other choice) change the tube but it's the first time I've done any real bicycle maintenance more advanced than installing lights and changing locks etc and my knowledge is very limited and all I've done has been through quick guides and intuition.

 

I changed the tube and all that just fine but when I spin the wheel it doesn't look straight but it's a new bike so there isn't any notable wear and tear. Is there any chance of installing the wheel 'incorrectly'. I'm speaking primarily in regards to safety. I don't want the wheel falling off or anything of the sort. The wheel positioning appears similar to before I took it off to replace the tube but as I said it kinda 'warps' when it spins and I couldn't adjust the wheel to get it straight again.

 

It's been a while since I looked at it and it's been in the shed since but off the top of my head it went as follows: rear wheel put onto the frame and pulled slightly back on it to make sure it's positioned correctly and as far back as possible, put gearbox back on and tightened the bolts on both sides. Everything looks to be as it should but the wheel warps as if something's wrong.

 

I've thought of just taking it to the nearest bike shop and get them to look at it but I've also thought that "damn it I need to do this". 

So any experts who can tell me what's up and what I need to do?

[Elite14]

6 hours ago, Trixanity said:

-snip-

so what i think you mean is that the rim is not true, which for lower end bike's is fairly common as long it doesn't move to much it will be fine, but if you want to get it really straight watch this video  

[asus killer]

to install the wheel make sure the bike is not in the air, let it sit in the floor, pull the wheel up and the bike down.

 

i really think without looking that what happened is not that their is nothing wrong with the wheel but rather the tire is not correctly sited on the rim, remove all the air from the tire, make sure the tire is sited and put some air on it making sure that the tire is straight in the rim. It's a very usual n00b mistake