From the parking for the Longs Peak trailhead, begin hiking on the famous, well-worn and used Longs Peak Trail. The last time we were on this trail was 2007. At that time, there had been some major trail work on the first mile and a half of the trail with gravel having been brought in to raise the bed of the trail. Farther on however, the trail was still in bad need of repairs, especially as you get above timberline and the trail becomes entrenched in places. The trail initially heads WNW and there's a junction about a half mile up where you should stay left. In another mile, the trail comes to a series of a half dozen switchbacks, then crosses "Alpine Brook" which originates in the Jim's Grove area. Avoid taking the alternate trail to Jims' Grove (if it's still an open trail) and continue to tree line and the "Mills Moraine." The trail will come to an intersection. The right hand fork continues on to the "boulder field" and the Long's Peak route. The left fork takes hikers to Chasm Lake, at the foot of the east face "diamond" of Long's Peak. In 2007, there was a solar toilet nearby.
The intersection mentioned before is well-signed. Take the left fork and head for Chasm Lake, enjoying the view of Peacock Lake and the Longs Peak Diamond/East Face. The trail loses a little elevation on its way to the rebuilt Ranger's cabin, the original destroyed in 2003 by avalanche. From there, the real fun begins. There has been now for a number of years, a climbers trail that heads on toward the couloir on the south (left) side of the "Ship's Prow," the very prominent and rugged ridge that comes off the left side of the Longs east face. Initially the trail heads up across tundra but soon gets into the rocks and switchbacks its way up through all the rocks, boulders, etc. As you work your way up the couloir, you may begin to question the sanity of this route because at the head of the couloir are technical cliffs. The key is a ledge/ramp leading off to the left that provides a walking path through a cliff section. This will be at about the 12,800 foot level and before you reach the cliffs at the head of the couloir. Before reaching that ramp you will likely encounter some larger slabs and boulders that will require some light, 3rd class scrambling. Breathe a sigh of relief when you find the ramp. The width of this ledge/ramp varies in reports from 2 to 10 feet. We did not have a measuring tape at the time and 29 years later, memory does not serve us well, but we had no problem making our way along the ramp just by walking, even though there is some exposure. Also be aware that earlier in the season, this entire couloir may be a snow ascent so ice axe and crampons or micro spikes may be required.
Some reports mention a second ramp cutting back to the right about 100 feet beyond the first ramp. Others make no mention. Above here, you still have some elevation gain to reach the loft and plenty more talus to scramble over but the slope becomes more gentle. "The Loft" is a broad, flat saddle between Meeker and Longs. Make a mental note of where you come out at the Loft, You will need this for the way back down. It may appear tempting to head over to Longs from here, but out of view is a technical 200 foot notch inhibiting progress. A combined Meeker/Longs day while possible, would be quite a long day for most anyhow. Better to combine Meeker with Lady Washington.
From the Loft, head SE to gain another 400 feet in elevation to the summit ridge over boulder-strewn terrain. Once on the summit ridge, the summit is not too far away, the last section offering some slabs and solid rock as the ridge narrows. The true summit is a large block that allows one climber at a time. Any register is usually located below and away from this block. Mounting it is regarded as a 3rd class move. Others regard it as 4th class. On the north side, things plummet away on the north face of Meeker, which you had a good view of from the Chasm Lake trail. Watch you step!
Enjoy the expansive view of the Colorado eastern plains below then return as you came to Chasm Lake and the Longs Peak trail. A climb of Meeker is also possible from the south by taking the trail from the Wild Basin TH to Sandbeach Lake. From there you'll have to bushwhack to Hunters Creek, then head up the south ridge of Meeker. This would be a longer and more difficult route than the Loft route. It's also possible to do the SE ridge of Meeker from a trail up to Lookout Mtn. An approach from this direction will require navigating the knife-edge ridge of Meeker by the eastern, lower summit. For those looking for some training before attempting the Capitol Peak knife edge, this might be a good place to practice. There are some interesting reports about this ridge online.
Links to other information, routes & trip reports for this peak that may be helpful.