Translating a datasheet into a PCB design (TPS63001)
The dots under the chip are indeed vias, and most likely help conduct heat to the other side of the board - which should have a fairly large area to help dissipate the heat.  They'll only help if you solder the exposed pad down, though.  If you only solder the pads at the ends, it may not be enough.
The current carrying traces should be wider as shown in layout from the datasheet, and as explained in the text in the datasheet.
One of the examples shows Vina, EN, and PS tied together, and then connected to Vin through a 100 Ohm resistor.  You have those three tied together, and that appears to be a resistor that connects them to Vin, OK.  Pulling PS high disables the power save mode.


Those are the things that I (as a hobbyist) can see.  

Eagle should have some kind of rules for trace widths.  If you set the power carrying connections to the proper type in the schematic, then it should tell you (or force you) to use wider traces when laying out those connections.  I don't use Eagle, though, so I can't say for sure what it does or how to set things up - most programs have some features along those lines, though.