My hair's naturally red, and I have dyed it black before. Most people liked it, but it is a pretty harsh contrast against pale skin so if you're not a regular at rock or heavy metal bars and clubs then you might stand out a bit (unless you're into tanning).
That said, everyone liked it better when I dyed it pillarbox red, and better still when I bleached it blonde.
You will need to dye your eyebrows and eyelashes as well, but I find that specialist eyebrow dye isn't all that strong. In photos my eyebrows always looked ginger, so I started using the hair dye on my eyebrows as well with no negative effects (so far), but my eyelashes still needed to be dyed at least twice.
Red hair tends to be really thick (which you'll know if you've already tried dying your hair before), so don't be scared to leave the dye on a bit longer than normal.
Personally, I love red hair and pale skin and if it's dyed black it won't stand out so much, so you might get less attention from guys than you're used to.
Me, I had a lot more attention from girls when I dyed mine black but you can't really compare the two.
In my opinion, a lighter color like your link will probably look okay with your presumed skin tone if you're very light skin (if you're a tad reddish in skin tone I don't know if I'd go with the style in the picture).
A dark color might make you look a little morbid or sickly because the contrast could be overpowering.
Stay a ginger, not because of the color itself (though I prefer reds), it's more because as you continually dye your hair, you are slowly but surely killing it. You'd rather have red hair, than no hair at all
If you're going to dye it, don't do it black, the roots and maintenance will be horrendous and it's not worth it. If you're really that bothered about dying it, do it subtle get the blond one but using highlights and lo-lights and get it done professionally.