You should be able to get sound that way, if all you need from the computer is 2 channel sound. Many TVs will not pass a multichannel sound out its digital output if it is input via HDMI, even if it will pass multichannel sound that it gets via broadcast. (I know, that sounds stupid, but that is the way many TVs are.) You will need to consult the manual for your TV to see what it says about this.
On the other hand, if you need more inputs on the receiver, you can get an HDMI switch box, such as the one from Oppo:
http://www.oppodigital.com/hm31/
But I would advise you to consult your manual for your TV before you buy anything. And if you only need 2 channel sound from your computer, it should work (though, again, I recommend that you consult your TV manual before assuming anything).
As for where to order cables, I recommend looking at one of the following:
http://www.monoprice.com/
http://www.parts-express.com/
Most places charge way too much for HDMI cables, though you can buy overpriced cables at the links above, too, if you prefer to waste money over their less expensive cables.
If the sound from your computer will be 2 channel PCM, Dolby Digital and dts, if you have the right outputs on the computer, you could run a digital connection (either coaxial or optical) from the computer to the receiver for the sound, and that should cost less than adding an HDMI switch box. You then would be using the HDMI from the computer to the TV just for video.
As for the "Lifestyle" speakers, if you do some searching at this site, you will find many people who hate Bose speakers. Some are more rabid than others about this. I do not recommend them, but you should listen to them, and listen to other types of speakers, and decide what you want, not what I or anyone else would have. After all, you will live with the results, not me or anyone else.