Well let's throw some numbers at traveling at twice the speed of light and see if the numbers add up:

Say a star was 100 light years away from you. It emitted light in the year 1911, and the light arrived at your position in the year 2011. Just as the light reached you, you started traveling towards the star at 2c (twice the speed of light). As you traveled towards the star you would be encountering younger and younger light from the star that was emitted after the original light that emitted in 1911 that reached you when you started traveling.

When you get to within 50 light years of the star, you will have traveled for 25 years, and traveled a distance of 50 light years. So the year is 2036, and the light that hits you there was emitted 50 years ago, so the light left the star in the year 1986.

When you get to within 25 light years of the star, you will have traveled for 12.5 more years (37.5 total), and traveled a distance of 25 light years (75 light years total). So the year is 2048.5, and the light that hits you there was emitted 25 years ago, so the light left the star in the year 2023.5.

When you get to within 1 light year of the star, you will have traveled for 12 more years (49.5 total), and traveled a distance of 24 light years (99 light years total). So the year is 2060.5, and the light that hits you there was emitted 1 year ago, so the light left the star in the year 2059.5.

When you get to the star, you will have traveled for .5 more years (50 years total), and traveled a distance of 1 light year (100 light years total). So the year is 2061, which is 50 years later than when you left in 2011, because you traveled for 50 years.

Yup, the numbers add up. Just like clockwork!