There are 10 levels in the Handiness skill. The handiness skill is useful around the house, because your sim will be able to repair broken objects, upgrade them and even make objects out of wood and sell them for profit. Also to make sure to have the lot trait "Homey" active, as it will give a +50% speed boost to those learning the skill.
To start, either wait for something to break, buy the woodworking craft table or buy the handiness skill books from a bookshelf. By level 2, you will be able to make simple upgrades and by level 9 you will be able to upgrade everything that can be upgraded. At level 10, you will be able to mentor other sims in the skill. The achievement will pop after your sim reaches level 10.
The easiest way I got my sim up to level 10 was to buy the woodworking craft bench and get my sim up to about level 4 of handiness. Then I bought about 4000 simoleons worth of the upgrade parts (4000 in total between all four of the different parts.) and I started upgrading everything I could. After upgrading a shower, toilet, two sinks, refrigerator, coffee maker, tv, radio, and computer my sim reached level 10 after about 2 hours (real world) of playing.