Building it from scratch is difficult..
A newer laptop with an intel i3 processor and at least 4gb of ram would run it smooth, but i think those are a little over 300$.
Try searching Best Buy.
I would recommend going with the below build instead though because the A8 is more powerful than the Celeron and if you choose to add a graphics card later it will run even better games.
The only big downfall of these builds is the lack of storage space, but 160GB should get you by until you have enough for a 500GB drive. Both builds have little upgradability due to their low cost motherboards or in the case of the APU build, there is no point in upgrading to a slightly better processor. Because of this, I would suggest going with the more powerful build which would have to be the one with an AMD processor.
P.S. teckmaster you didn't include a case in your list of parts.
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If you end up getting an AMD CPU, they have slower cores, but more of them. But without Optifine Multicore (that gives you graphical bugs) Minecraft only uses 1 So i would reccomend going intel for a Gaming computer, AMD has its place, but not in gaming.
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A newer laptop with an intel i3 processor and at least 4gb of ram would run it smooth, but i think those are a little over 300$.
Try searching Best Buy.
Edit: Some thing like this: http://www.bestbuy.com/site/Acer+-+11.6%26%2334%3B+Aspire+One+Laptop+-+2GB+Memory+-+320GB+Hard+Drive+-+Ash+Black/5688602.p?id=1218684536310&skuId=5688602&st=%205688602&cp=1&lp=1
Spend the extra money on RAM.
CPU: AMD A8-3870K 3.0GHz Quad-Core Processor ($114.98 @ NCIX US)
Motherboard: MSI A75A-G35 ATX FM1 Motherboard ($69.99 @ Newegg)
Memory: Mushkin Essentials 4GB (1 x 4GB) DDR3-1333 Memory ($19.98 @ Outlet PC)
Storage: Seagate Momentu 500GB 2.5" 7200RPM Internal Hard Drive ($56.98 @ NCIX US)
Power Supply: Cooler Master 400W ATX12V Power Supply ($28.99 @ SuperBiiz)
Optical Drive: LG GH24NS90 DVD/CD Writer ($16.99 @ Newegg)
Total: $307.91
http://pcpartpicker.com/p/dECc
Wow, most if not all of the above replies are horrible.
CPU: Intel Celeron G540 2.5GHz Dual-Core Processor ($44.99 @ Amazon)
Motherboard: MSI H61M-P31 (G3) Micro ATX LGA1155 Motherboard ($49.54 @ NCIX US)
Memory: Mushkin Essentials 4GB (1 x 4GB) DDR3-1333 Memory ($19.98 @ Outlet PC)
Storage: Western Digital Scorpio Black 160GB 2.5" 7200RPM Internal Hard Drive ($48.21 @ NCIX US)
Video Card: Gigabyte Radeon HD 6570 1GB Video Card ($59.98 @ NCIX US)
Case: NZXT Source 210 (Black) ATX Mid Tower Case ($34.99 @ Newegg)
Power Supply: Corsair 430W ATX12V Power Supply ($42.99 @ NCIX US)
Optical Drive: LG GH24NS90 DVD/CD Writer ($17.99 @ Amazon)
Total: $318.67
http://pcpartpicker.com/p/dEXO
I would recommend going with the below build instead though because the A8 is more powerful than the Celeron and if you choose to add a graphics card later it will run even better games.
CPU: AMD A8-3850 2.9GHz Quad-Core Processor ($99.97 @ CompUSA)
Motherboard: Asus F1A55-M LE Micro ATX FM1 Motherboard ($59.98 @ NCIX US)
Memory: Mushkin Essentials 4GB (1 x 4GB) DDR3-1333 Memory ($19.98 @ Outlet PC)
Storage: Western Digital Scorpio Black 160GB 2.5" 7200RPM Internal Hard Drive ($48.21 @ NCIX US)
Case: NZXT Source 210 (Black) ATX Mid Tower Case ($34.99 @ Newegg)
Power Supply: Corsair 430W ATX12V Power Supply ($42.99 @ NCIX US)
Optical Drive: LG GH24NS90 DVD/CD Writer ($17.99 @ Amazon)
Total: $324.11
http://pcpartpicker.com/p/dEZe
The only big downfall of these builds is the lack of storage space, but 160GB should get you by until you have enough for a 500GB drive. Both builds have little upgradability due to their low cost motherboards or in the case of the APU build, there is no point in upgrading to a slightly better processor. Because of this, I would suggest going with the more powerful build which would have to be the one with an AMD processor.
P.S. teckmaster you didn't include a case in your list of parts.