Hi, here is my three cents.
I used to own the Linn Majik and Classik with CD built in to the unit. You can do much better!!! Even on your small budget.
I like alot of "presence" in my music, "You are there" feeling. And as such, I have found the Jolida hybrid tube / ss integrated amps to be excellent! (Can usually get a good deal on them here at the AGon.) Make sure you have effecient speakers, as the amp section is not overly powerful, but better suited for speakers of 88db and up. 90 db would be even better.
After doing some in home "ABing", I found that my very affordable DVD player from Panasonic sounded better than the Linn CD player built into the Classik I had a year ago. Ouch! Which is what got me rather disenchanted, and started looking elsewhere for the whole setup. Linn is a very reputable company, but I found that even after listening to their higher priced gear is not my "cup of tea".
Some great affordable name brands to check out for gear are Jolida, Music Hall. By the way, you might want to visit with Walter Liederman at Underwood Hi-Fi a bit, or some other guys who advertise on the AGon for this gear. For myself, I can attest that Walter is a straight shooter, will listen to your sonic thoughts, and is very respectful and patient of any final decisions you make. I wish all my dealer experiences in 20 years of HiFi-ing were as good.
By the way, I like the Jolida CD player too. But based on the tubes you run in the integrated, you may prefer the sound of another CD player.
I do not know anything about your computer hook up. You will have to get that integrated on your own.
One of the nicest things about all this gear is that it can be moddified and improved upon. It makes a big difference, and is an excellent return on the investment. Can grow with the piece of gear.
Good Luck, write if questions.
Rob