That Beats Audio stuff, say, on a windows laptop is the worst. (the software stuff that is what we get nowadays- there was an initial hardware implementation, but that was the first gen products)
From what I have seen when trying to disable it, they have mucked with the software bass and treble controls in multiple places throughout the system,.. some turning them fully up, others turning them fully down,.. so that if you 'turn off the beats audio' the sound will be horrid, and only if you persevere for a bit longer will you find all the places needed to return the laptop to flat sound. Makes sense cause when showing friends how great the 'beats' laptop is, and disabling the feature to show how crap it sounds without it,.. well it does become crap when you click the button in only one place. If we alter all three places that changes are made on a 'beats' laptop (software) then yes, we have nice sound again,.. but I digress..
HTC phone is Android, and it is known for good sound. But I cannot with any certaintly, based off what beats software does on a windows PC platform, trust that the phone is clear.
I guess this is in reply to the "beats' issue.
Regarding your laptop, sure it has a low cost DAC, but that is the least of the worries. Laptops have a horrible electrical environment. Some have a reconfiguring headphone jack that can double as an optical out. I like this idea most as we know that we are free of electrical polution of the signal. I do think as a secondary option a USB external DAC would also cut it.
Perhaps if you budget the benefits of an amp after a decent DAC, and buy an all-in-one, the increased price point will get you a better part. Just know if you are trying to do this on the cheap, and a certain amount of any item has to go on packaging and casing and profit, then I reckon by buying an all-in-one (in the budget space), a lil research as to which to buy might see you sail into some pretty sweet territory.
As I live in Aussieland, we have such a different range of parts/ and prices/ it would be impossible to recommend.
I did recently sell a musiland MD10 dac/headphone amp for around $100 (Aus dollars) and I do believe that second hand is the place to go when on a tight budget as, these parts don't really wear out, so much as wear-in.
Glad to see you realise that you can get soo much more from your headphones.. success with the project!